Post on 02-Aug-2020
PERSISTENT SCATTERER INTERFEROMETRY IN
NATURAL TERRAIN
a dissertation
submitted to the department of electrical engineering
and the committee on graduate studies
of stanford university
in partial fulfillment of the requirements
for the degree of
doctor of philosophy
Piyush Shanker Agram
August 2010
http://creativecommons.org/licenses/by-nc/3.0/us/
This dissertation is online at: http://purl.stanford.edu/fm943vt7275
© 2010 by Piyush Shanker Agram. All Rights Reserved.
Re-distributed by Stanford University under license with the author.
This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.
ii
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Howard Zebker, Primary Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
George Hilley
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Paul Segall
Approved for the Stanford University Committee on Graduate Studies.
Patricia J. Gumport, Vice Provost Graduate Education
This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.
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Abstract
Time-series InSAR techniques are designed to estimate the temporal characteristics
of surface deformation by combining information from multiple SAR images acquired
over time. In many cases, these techniques also enable us to measure deformation
signals in locations where conventional InSAR fails and also to reduce the error
associated with deformation measurements. Among these techniques, Persistent
Scatterer (PS) methods work by identifying the ground resolution elements that are
dominated by a single scatterer. A persistent scatterer exhibits reduced baseline
and temporal decorrelation due to its stable, point-like scattering mechanism. In PS
analysis, a set of interferograms formed with a single master scene are processed at
single look resolution in order to maximize the signal-to-clutter ratio (SCR) of the
resolution elements containing a single dominant scatterer. In urban terrain, buildings
and other man-made structures often act as PS due to their corner reflector-like
scattering behavior and high radar reflectivity. Hence, traditional SAR amplitude-
based PS-InSAR techniques have proved to be very effective in urban terrain. In
natural terrain, the absence of bright manmade structures makes reliable estimation
of deformation using PS-InSAR techniques a challenging task. The main obstacle is
in the phase unwrapping stage, where the solutions are directly dependent on the PS
network density.
We have developed a two pronged approach to improve the applicability of PS-
InSAR techniques to natural terrain - increasing PS network density and improving
the reliability of phase unwrapping algorithms. We first present an information
theoretic approach to PS pixel selection and demonstrate the ability of these new
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algorithms in identifying a denser network of PS in natural terrain. We then address
the spatio-temporal (three dimensional) phase unwrapping problem applicable to
sparse and non-uniformly sampled time-series InSAR data sets, and present two novel
phase unwrapping algorithms. We demonstrate the efficacy of our new PS selection
technique with experimental results from the San Francisco Bay Area, Lyngen region
of Norway and the creeping section of the Central San Andreas Fault. We explain the
salient features of our new “edgelist” phase unwrapping algorithm with results from
the Central San Andreas Fault region north of Parkfield, CA. We provide detailed
comparisons of the estimated line of sight velocity and deformation time-series with
results from other time-series InSAR algorithms developed by research groups based
in NORUT, Norway and IREA-CNR, Italy.
iii
Acknowledgments
Piyush Shanker Agram
Stanford, CaliforniaAugust 12, 2010
I owe my deepest gratitude to my advisor, Prof. Howard Zebker, for his untiring
guidance, advice and mentorship, and for being a great teacher and a source of
inspiration. I’m also grateful to Prof. Paul Segall for his encouragement, guidance
and insightful research advice. I’m indebted to George Hilley for kindly agreeing to
be the third thesis reader and providing valuable comments that helped improve this
thesis.
I am indebted to Andrew Hooper for being available to answer all my questions
patiently during the early stage of my Ph.D and for constructive suggestions later on.
I also owe the completion of this thesis to ideas and projects executed in collaboration
with Tom Rune Lauknes, Francesco Casu, Isabelle Ryder and Mariana Eneva. Other
colleagues who have been instrumental in my education are Leif Harcke, Noa Bechor,
Fayaz Onn, Ana Bertran Ortiz, Shadi Oveisgharan, Lauren Wye, Cody Wortham,
Albert Chen, Jessica Reeves and Jingyi Chen.
My work was supported generously by the NASA Earth and Space Science
Fellowship (NESSF) and other grants from the USGS, NSF and NASA. I would also
like to thank Prof. Leonard Tyler and Prof. Jerry Harris for supporting my education
by extending research and teaching assistantships.
I’m forever indebted to my parents, my sister, my uncle Ravi Nanjundiah
and family for their constant support, encouragement and guidance. A number
of friends have been instrumental in making my stay at Stanford a joyful and
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memorable experience, including Sachin Premasuthan, Ankit Bhagatwala, Stephanie
Hurder, Mamta Parakh, Tanmay Chaturvedi, Debarun Bhattacharjya, Sameer
Parakh, Shankar Swaminathan and Hari Kannan. My interaction with the Bechtel
International Center, EV Community Associate Program and the Stanford Cricket
Club also provided a refreshingly relaxing break from research and many memorable
experiences that I will fondly cherish.
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Contents
Abstract ii
Acknowledgments iv
1 Introduction 1
1.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Thesis Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 InSAR : Background 6
2.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Synthetic Aperture Radar . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 SAR Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Limitations of conventional InSAR . . . . . . . . . . . . . . . . . . . 12
2.5 Multi-temporal InSAR . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Math models 16
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Interferometric phase components . . . . . . . . . . . . . . . . . . . . 18
3.2.1 Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.2 Geometric phase component (Δφε) . . . . . . . . . . . . . . . 19
3.2.3 Atmospheric phase screen . . . . . . . . . . . . . . . . . . . . 22
3.2.4 Orbital errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.5 Other phase terms . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Scattering Signal Models . . . . . . . . . . . . . . . . . . . . . . . . . 25
vi
3.3.1 Constant Signal Model . . . . . . . . . . . . . . . . . . . . . . 25
3.3.2 Gaussian signal model . . . . . . . . . . . . . . . . . . . . . . 27
3.4 StaMPS framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4.1 Preliminary PS candidate selection . . . . . . . . . . . . . . . 31
3.4.2 Weighted estimation of correlated phase terms and DEM error 31
3.4.3 PS selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4.4 Sidelobe effects . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4.5 Other PS frameworks . . . . . . . . . . . . . . . . . . . . . . . 34
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Persistent Scatterer Selection 36
4.1 Amplitude based PS Selection . . . . . . . . . . . . . . . . . . . . . . 36
4.1.1 Amplitude Dispersion . . . . . . . . . . . . . . . . . . . . . . . 36
4.1.2 Signal-to-Clutter Ratio . . . . . . . . . . . . . . . . . . . . . . 38
4.1.3 Scripps technique . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Elimination using a temporal model . . . . . . . . . . . . . . . . . . . 40
4.3 StaMPS PS selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4 Maximum Likelihood PS selection . . . . . . . . . . . . . . . . . . . . 43
4.4.1 Estimator properties . . . . . . . . . . . . . . . . . . . . . . . 44
4.4.2 Threshold Selection . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5 An example application . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5 Phase Unwrapping 54
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Residue-based unwrapping techniques . . . . . . . . . . . . . . . . . . 57
5.2.1 Network programming approach . . . . . . . . . . . . . . . . . 58
5.3 Reduction of sparse data on a regular grid . . . . . . . . . . . . . . . 59
5.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.3.2 Geometrical reduction to regular problem . . . . . . . . . . . . 62
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5.3.3 Branch cuts, flows and cost functions . . . . . . . . . . . . . . 65
5.3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.4 Three-dimensional unwrapping . . . . . . . . . . . . . . . . . . . . . . 69
5.5 Edgelist phase unwrapping formulation . . . . . . . . . . . . . . . . . 70
5.5.1 Total unimodularity . . . . . . . . . . . . . . . . . . . . . . . 74
5.5.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.6 Incorporating other geodetic measurements as constraints . . . . . . . 77
5.6.1 Total unimodularity . . . . . . . . . . . . . . . . . . . . . . . 79
5.7 Case study: Creeping section of the San Andreas Fault . . . . . . . . 80
5.8 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 86
6 Rockslide mapping 88
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Study area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.3 Available SAR data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.3.1 Atmospheric phase correlated with topography . . . . . . . . . 94
6.4 Short description of the SBAS algorithm . . . . . . . . . . . . . . . . 97
6.4.1 Estimation of DEM error and low pass displacement signal . . 98
6.4.2 Estimation of cumulative phase . . . . . . . . . . . . . . . . . 99
6.4.3 Atmospheric filtering and final displacement time series estima-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.5.1 PS-InSAR procedure . . . . . . . . . . . . . . . . . . . . . . . 101
6.5.2 SBAS procedure . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.5.3 Differences in implementation . . . . . . . . . . . . . . . . . . 103
6.5.4 Final displacement estimates . . . . . . . . . . . . . . . . . . . 104
6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.6.1 Gamanjunni . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.6.2 Rismmalcohkka . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.6.3 Nordnes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
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6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7 Noise in PS-InSAR 114
7.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.2.1 Tandem SAR scenes . . . . . . . . . . . . . . . . . . . . . . . 118
7.2.2 Inter-comparison with SBAS . . . . . . . . . . . . . . . . . . . 118
7.2.3 Comparison with alignment arrays . . . . . . . . . . . . . . . 119
7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8 Summary 128
8.1 Future Work and Improvements . . . . . . . . . . . . . . . . . . . . . 129
8.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
References 131
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List of Tables
3.1 Spectral characteristics for various phase components of the observed
interferometric phase for a PS pixel (Hooper, 2006). . . . . . . . . . . 30
4.1 Comparison between different PS selection methods for two test regions
in the San Francisco Bay Area . . . . . . . . . . . . . . . . . . . . . . 52
5.1 Parameters needed to represent a regular grid 2D unwrapping problem
of size M x N pixels using both the edgelist and MCF formulations. . 73
6.1 Used SAR scenes in the rockslide data set. . . . . . . . . . . . . . . . 96
6.2 Main differences between SBAS and PS. . . . . . . . . . . . . . . . . 112
6.3 Processing results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.1 Statistics of pixels identified by PS and SBAS techniques. . . . . . . . 120
7.2 LOS time-series differences for tandem dates in the PS data set. . . . 120
7.3 Differences between PS and SBAS estimates for LOS velocity and time-
series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.4 Difference between PS-InSAR time-series and alignment array creep
estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
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List of Figures
2.1 SAR imaging geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Real and synthetic aperture. . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 SAR baseline geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Example interferogram. . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Scattering Mechanisms. . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1 Geometric phase component and perpendicular baseline. . . . . . . . 20
3.2 Probability distributions of statistics for constant signal model. . . . . 28
3.3 Probability distributions of statistics for Gaussian signal model. . . . 29
3.4 Flowchart of the StaMPS framework. . . . . . . . . . . . . . . . . . . 33
4.1 Amplitude dispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Signal-to-Clutter ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 MLPS estimator vs conventional coherence estimator. . . . . . . . . . 45
4.4 Effect of sample size on MLPS estimator performance. . . . . . . . . 46
4.5 Estimated SCR vs true SCR for simulated phase sequences. . . . . . 47
4.6 Determining threshold using standard deviation of phase. . . . . . . . 48
4.7 RPAR as a function of SCR thresholds. . . . . . . . . . . . . . . . . . 49
4.8 Comparison of PS mask. . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.9 Comparison of PS identification algorithms. . . . . . . . . . . . . . . 52
5.1 Residues in 2D interferograms. . . . . . . . . . . . . . . . . . . . . . . 58
5.2 Reduction of a sample sparse data set to a regular 2D grid. . . . . . . 61
5.3 Types of residues in interpolated data sets. . . . . . . . . . . . . . . . 63
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5.4 Branch cuts in the original sparse and interpolated regular datasets. . 66
5.5 Application of the interpolation algorithm to simulated data. . . . . . 68
5.6 Application of the interpolation algorithm to the Bam Earthquake. . 69
5.7 Edgelist formulation vs MCF formulation. . . . . . . . . . . . . . . . 73
5.8 Example data set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.9 Edgelist formulation corresponding to example data set. . . . . . . . 78
5.10 Map of the Central San Andreas fault area. . . . . . . . . . . . . . . 80
5.11 Phase unwrapping model for the Central San Andreas data set. . . . 81
5.12 Edgelist unwrapping results for the Central San Andreas data set. . . 82
5.13 Cumulative time-series of creep across the Central San Andreas Fault. 85
5.14 Stepwise 3D phase unwrapping results for Central San Andreas data set. 86
6.1 Photomontage of rockslide sites and possible scattering mechanisms. . 91
6.2 Map of study area in Troms county. . . . . . . . . . . . . . . . . . . . 93
6.3 Baseline plot for the PS and SBAS data sets. . . . . . . . . . . . . . . 94
6.4 Topography-related atmospheric propagation delay. . . . . . . . . . . 95
6.5 Detailed results from the Gamanjunni slide. . . . . . . . . . . . . . . 104
6.6 Detailed results from the Rismmalcohkka slide. . . . . . . . . . . . . 105
6.7 Detailed results from the Nordnes slide. . . . . . . . . . . . . . . . . . 106
7.1 Map of the areas in San Francisco Bay area used for PS-SBAS
comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.2 Baseline plot for the PS and SBAS data sets. . . . . . . . . . . . . . . 117
7.3 Histogram of tandem data LOS time-series differences in PS data set. 121
7.4 Comparison LOS velocities estimated using PS and SBAS. . . . . . . 123
7.5 LOS time-series for select stations estimated using both PS and SBAS. 124
7.6 Histogram of differences between LOS velocity and time-series esti-
mates using PS and SBAS techniques. . . . . . . . . . . . . . . . . . 124
7.7 PS-InSAR time-series vs alignment array creep estimates. . . . . . . . 125
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List of Abbreviations
1-D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . one dimensional
2-D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .two dimensional
3-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three dimensional
N -D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .N dimensional
APS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atmospheric Phase Screen
DEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Digital Elevation Model
DLR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . German Aerospace Center
ESA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .European Space Agency
Envisat . . . . . . . . . . . . . . . . . . . . . . . . . . European Space Agency’s environmental satellite
ERS-1 . . . . . . . . . . . . . . . . . European Space Agency’s Earth Remote Sensing satellite 1
ERS-2 . . . . . . . . . . . . . . . . . European Space Agency’s Earth Remote Sensing satellite 2
ER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error Rate
FFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Fast Fourier Transform
GNSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Global Navigation Satellite System
GPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Global Positioning System
GSAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generic SAR processor from Norut
IFG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interferogram
JPL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NASA’s Jet Propulsion Laboratory
LOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Line-of-sight
LP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Linear Program
MCF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Minimum Cost Flow
xiii
ML. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Maximum Likelihood
MST. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Minimum Spanning Tree
NASA . . . . . . . . . . . . . . . . . . . . . . . . . . . . National Aeronautics and Space Administration
NGU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geological Survey of Norway
Norut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Northern Research Institute, Norway
Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RAdio Detection and Ranging
SAR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Synthetic Aperture Radar
InSAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interferometric Synthetic Aperture Radar
IREA-CNR. . . . . . . . . .Institute for Electromagnetic Sensing of Environment, Napoli
D-InSAR. . . . . . . . . . . . . . . . . . . .Differential Interferometric Synthetic Aperture Radar
PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probability Distribution Function
PS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Persistent Scatterers or Permanent Scatterers
PS-InSAR . . . . . . . . . . Persistent Scatterer Interferometric Synthetic Aperture Radar
PSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Persistent Scatterer Interferometry
RMSE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Root Mean Squared Error
RPAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random Pixels Acceptance Rate
SB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Short Baseline
SBAS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Small Baseline Subset Algorithm
SEASAT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .NASA’s remote sensing satellite
SCR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal-to-clutter ratio
SNR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Signal-to-noise ratio
SLC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single look complex SAR image
SRTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shuttle Radar Topography Mission
StaMPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stanford Method for PS
SFSU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . San Francisco State University
TUM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Unimodularity
USGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . United States Geological Survey
xiv
WGS-84 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . World Geodetic System 1984
ZWD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zenith Wet Delay
xv
List of Symbols
Δφ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interferometric phase
Δρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Difference in LOS range
λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wavelength
φbase . . . . . . . . . . . . . . . . . . . . . . . Interferometric phase due to perfectly ellipsoidal Earth
φtopo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interferometric phase due to topography
φdefo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Interferometric phase due to deformation
φatm . . . . . . . . . . . . . . . . . . . . . . . . Interferometric phase due to atmospheric phase screen
φnoise, φn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interferometric phase noise
B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interferometric baseline
B⊥ . . . . . . . . . . . . . . . . . . Component of Interferometric baseline perpendicular to LOS
α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baseline angle with the horizontal
θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Look angle
Δφε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase error due to sub-pixel position of PS
Δφorb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase error due to orbit errors
Δh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error in DEM
ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sub-pixel position in range of dominant scatterer
Δr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Range resolution
η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Sub-pixel position in azimuth of dominant scatterer
v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Relative velocity of the SAR imaging platform
FDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Doppler centroid of a SAR scene
zi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex return from pixel i
xvi
fθ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .PDF of SAR pixel phase
fφ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PDF of InSAR pixel phase
fr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PDF of SAR pixel amplitude
μn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Mean of random variable n
σn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard deviation of random variable n
γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal-to-Clutter Ratio
ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlation
ρtemp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Temporal coherence
ρthr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlation threshold
Pobs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Probability as computed using observed data
Psim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Probability as computed using simulations
γML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximum likelihood estimate of γ
Cij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Cost of unit flow on edge (i, j)
Kij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integer flow on edge (i, j)
Pij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integer flow from i to j on edge (i, j)
Qij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integer flow from j to i on edge (i, j)
S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Delaunay triangulation of set of sparse points S
S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Discrete Voronoi diagram of set of sparse points S
ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integer cycle multiplier for pixel i
ψi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wrapped phase observation for pixel i
φi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unwrapped phase observation for pixel i
v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean velocity
a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean acceleration
Δa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Mean acceleration variation
xvii
Chapter 1
Introduction
Surface deformation measurements are critical for the study and detailed understand-
ing of of tectonics, earthquakes, volcanism and landslides. Geodetic measurements
provide invaluable information on the accumulation or release of strain in seismically
active faults or stretching of the crust as a result of magma migration from the mantle
through the crust in volcanoes. Geodetic measurements also play an important role
in monitoring active landslides and provide important insight into their kinematics.
Global positioning system (GPS) networks and interferometric synthetic aperture
radar (InSAR) are the two most popular techniques used for measuring surface
deformation. InSAR has an unique advantage in its ability to measure deformation
with centimeter scale accuracy over large contiguous areas (few hundred kilometers).
However, almost any interferogram includes large areas where the signals decorrelate
due to the imaging geometry, vegetation or change in surface scattering properties and
no reliable measurement is possible. Moreover the deformation measurements due to a
single interferogram are affected by the variation in atmospheric properties. Persistent
scatterer (PS) InSAR refers to a family of time-series InSAR techniques that addresses
both the decorrelation and atmospheric problems of conventional InSAR.
Early PS-InSAR techniques were designed to identify very bright scatterers, whose
scattering properties vary little with imaging geometry and time. Such techniques
work well in urban areas where man-made structures behave like strongly reflecting
corner reflectors but fail to perform satisfactorily in vegetated terrain. The StaMPS
1
CHAPTER 1. INTRODUCTION 2
(Stanford Method for Persistent Scatterers) technique, developed by Andy Hooper,
was the first PS-InSAR method designed to estimate a deformation signal even in the
absence of very bright scatterers using a self-consistent network of PS. In this work,
we build upon StaMPS PS selection framework and develop an information theoretic
approach to improve PS selection in non-urban and vegetated terrain. We apply our
new Maximum Likelihood (ML) PS selection technique to identify a denser network of
PS in non-urban regions of the San Francisco Bay Area. We then present a new edge-
based phase unwrapping algorithm that is more flexible than the conventional network
programming phase unwrapping techniques. We apply this algorithm to unwrap a
sparse PS dataset and estimate the time-dependent creep across the Central San
Andreas Fault. This constitutes the first such analysis of this heavily decorrelated
section of the San Andreas Fault using time-series InSAR techniques. We then
compare the results of our new PS selection techniques against those obtained with
two different short baseline time-series InSAR implementations. We compare PS-
InSAR results and small baseline subset algorithm (SBAS) results from Northern
Research Institute, Tromso (Norway) over the rockslides in Lyngen region of Norway.
We also compare PS-InSAR and SBAS results from IREA-CNR, Naples (Italy) on
a pixel-by-pixel basis in the radar coordinates to characerize the noise properties of
time-series InSAR deformation estimates. Significantly, we observe that that these
results agree to within 1 mm/yr and 5 mm line of sight (LOS) velocity and absolute
displacement respectively.
1.1 Contributions
There are three aspects to this dissertation. First, we describe a new Bayesian PS
selection technique based on statistical models that extends the applicability of PS-
InSAR techniques in natural terrain. Second, we describe new phase unwrapping
algorithms for sparse PS networks that significantly improves the accuracy of
deformation estimates. We then apply these improved PS-InSAR techniques to study
CHAPTER 1. INTRODUCTION 3
numerous geophysical phenomenon. Finally, we compare the performance of our
newly developed PS-InSAR techniques against short baseline (SB) techniques from
independent research groups in Europe and characterize the noise properties of our
time-series results. Below we summarize the main contributions of this work:
1. We design an information theoretic approach to identify persistent scattering
pixels in a series of interferograms that outperforms other published algorithms
in natural terrain.
2. We develop a new framework that reduces a sparse two-dimensional phase
unwrapping problem to a regular two-dimensional phase unwrapping problem.
3. We develop a new generic phase unwrapping formulation capable of handling
multidimensional datasets and incorporating external geodetic measurements
such as leveling surverys and GPS as constraints.
4. We apply our new methods to estimate the time-series for creep across the
central Hayward fault and deformation of areas along the Bay in San Mateo
county and Alameda county of California.
5. We apply our new PS selection method and phase unwrapping techniques to
studying time-dependent creep across the central section of the San Andreas
Fault.
6. We provide a detailed one-to-one comparison of deformation time-series esti-
mated using PS-InSAR and SBAS algorithms for slow moving landslides in
Lyngen region, Norway.
7. We conduct a pixel-by-pixel comparison of PS-InSAR and SBAS results in the
San Francisco Bay Area and estimate the noise levels in our PS-InSAR estimates
to be 1 mm/yr in LOS velocity and 5 mm in absolute LOS displacement
compared to the SB estimates.
CHAPTER 1. INTRODUCTION 4
8. We compare our PS-InSAR deformation time-series with creep measurements
from the SFSU-USGS alignment arrays along the Hayward Fault and show that
the creep estimates agree to within 1.5 mm LOS displacement and 0.5 mm/yr
LOS velocity.
1.2 Thesis Roadmap
In Chapter 2 we provide a brief overview of InSAR and in Chapter 3 present a detailed
overview of mathematical models used to describe the statistical behavior of persistent
scatterers in interferograms. This chapter includes statistical distributions for
amplitude and interferometric phase for signal models commonly used for describing
PS.
In Chapter 4 we review PS selection techniques in literature so far and introduces
the bayesian Maximum Likelihood (ML) PS selection algorithm. We apply the ML
PS technique to two regions in the San Francisco Area and show that the algorithm
identifies a denser network of PS in non-urban vegetated terrain than other published
algorithms. This PS selection technique has been published in Geophysical Research
Letters (Shanker and Zebker, 2007).
In Chapter 5 we describe two new phase unwrapping algorithms for unwrapping
sparse data. The first technique involves a simple reduction of a sparse two-
dimensional phase unwrapping problem into a regularly sample two-dimensional phase
unwrapping problem that can be solved with any generic phase unwrapping algorithm.
This method has been published in IEEE Geoscience and Remote Sensing Letters
(Shanker and Zebker, 2008). Next, we describe a new linear programming formulation
that relies on the edges of an unwrapping grid as a basic construct. This unwrapping
formulation is more flexible than conventional network programming approaches and
can incorporate additional geodetic information as constraints in the unwrapping
process. We apply this unwrapping technique to estimate the time-dependent creep
CHAPTER 1. INTRODUCTION 5
across the Central San Andreas Fault. This technique has been published in the
Journal of Optical Society of America A (Shanker and Zebker, 2010).
In Chapters 6 and 7 we compare the performance of our PS-InSAR algorithms
with the SBAS algorithm. In Chapter 6, we apply our techniques to study rockslides
in Lyngen region of Norway and present a one-to-one comparison with SBAS
results processed at Northern Research Institute (NORUT), Norway. This detailed
comparison has been published in Remote Sensing of the Environment (Lauknes et al.,
2010). In Chapter 7 we present a pixel-by-pixel comparison of our PS-InSAR results
and SBAS results from IREA-CNR, Italy. We also compare our PS-InSAR estimates
against creep measurements from alignment arrays along the Hayward Fault. A
paper describing this comparison in detail has been submitted to IEEE Geoscience
and Remote Sensing Letters.
Finally, in Chapter 8 we provide a dissertation summary and suggestions for future
work.
Chapter 2
InSAR : Background
Persistent Scatterer Synthetic Aperture Radar Interferometry (PS-InSAR) is a family
of extensions to the Interferometric Synthetic Aperture Radar (InSAR) technique,
that allow us to estimate a deformation time-series for regions that are traditionally
considered decorrelated under conventional InSAR techniques. In this chapter, we
review the traditional InSAR technique for studying ground deformation and its
associated shortcomings. We also review multi-temporal InSAR methods that have
been developed to address the shortcomings of standard InSAR and discuss their
strengths and weaknesses.
2.1 History
The word “radar” itself is an acronym for radio detection and ranging. Radar systems
were initially developed in the first half of the 20th century to determine the position or
course of a moving object like an ocean-going vessel or an airplane. Pulse compression
signal processing techniques have been traditionally used to improve the signal to
noise ration and then locate targets with an error of few meters (Cumming and
Wong, 2005; Soumekh, 1999). Synthetic Aperture Radar (SAR) is an extension of
the mapping radar that creates a highly directional beam using sophisticated signal
processing to combine information from multiple echoes to accurately estimate the
azimuth position of the target as well. The synthetic aperture concepts enable us to
6
CHAPTER 2. INSAR : BACKGROUND 7
image with a resolution on the order of meters with relatively small physical antennas.
As a result, SAR is an excellent tool for high resolution remote sensing of the Earth’s
surface from space. After experimenting with airborne SAR instruments in the 60s
and 70s, NASA launched the first space-borne SAR satellite SEASAT in 1978 for
ocean studies (Fu and Holt, 1982).
2.2 Synthetic Aperture Radar
Airborne and spaceborne SAR systems typically have a fixed side-looking antenna
that illuminates a strip or swath parallel to the sensor’s ground track with a series of
monochromatic microwave pulses (Figure 2.1). The platform’s flight direction is called
the azimuth direction and the direction of the main lobe of the transmitting antenna
is called the range direction. The antenna, when inactive between transmission of
pulses, is designed to receive the scattered echoes from the illuminated surface of
earlier transmitted pulses. From antenna theory, the area illuminated on the ground
is inversely proportional to the physical shape and dimensions of the antenna (Skolnik,
2001). Therefore, to obtain fine azimuth resolution in real-aperture radar systems we
would need a very long antenna.
SAR is an alternative solution to using a long physical antenna. The SAR
concept separates two targets at the same range but different azimuth positions by
their different relative velocities with respect to the moving platform. The reflected
monochromatic waves from two different scatterers in the same illuminated beam
have different Doppler shifts or phases associated with them. Using the knowledge of
the path of the imaging platform, we can compute the exact phase history for every
point target on the ground. We combine information from multiple echoes, effectively
creating a synthetic longer aperture (Figure 2.2) to separate targets within the same
illuminated beam. Resolution is dependent on the total amount of phase information
available for each target. The longer a target is illuminated, the better is our ability
to resolve it. Thus, SAR enables us to create high resolution images using small
CHAPTER 2. INSAR : BACKGROUND 8
Figure 2.1: SAR imaging geometry
physical antennas. There are numerous published algorithms (Cumming and Wong,
2005; Soumekh, 1999) that can be used to process the recorded echoes to create high
resolution images using the SAR concept.
The imaging geometry shown in Figure 2.1 is called the stripmap mode and
is the most common. Modern phased-array antennas implement complicated data
acquisition strategies, e.g, ScanSAR and spotlight SAR, to increase the area imaged
by the radar platform. The output from SAR processing algorithm is a single look
complex (SLC) image, a two dimensional array of complex numbers, representing the
brightness and phase of the scatterers on the ground. The indices of pixels in the
matrix are directly related to the azimuth and range position of the scatterers with
respect to a reference point on the platform’s path.
CHAPTER 2. INSAR : BACKGROUND 9
Figure 2.2: (a) Real aperture antenna, (b) Synthetic aperture antenna created bycombining information from many pulses and (c) a real aperture antenna that isequivalent to the synthetic aperture antenna.
2.3 SAR Interferometry
Synthetic aperture radar interferometry (InSAR) exemplifies multiplicative inter-
ference. The almost monochromatic nature of the scattered echoes enables us to
combine the phase information from two or more SAR images acquired over the
same area either simultaneously or at different times. The two SAR images are
generally acquired from slightly different imaging geometries. The second SLC must
be precisely coregistered and resampled to the geometry of the first SLC (Zebker and
Goldstein, 1986; Sansosti et al., 2006). The interferometric phase is then computed
by multiplying the first SLC with the complex conjugate of the coregistered second
SLC. The resulting complex valued image is called an interferogram (IFG). Ignoring
any time delays in the imaging hardware, the estimated interferometric phase can be
directly related to the difference in path length to a target from the imaging platform
in the line of sight (LOS) direction (Figure 2.3).
Δφ = −4π
λΔρ (2.1)
CHAPTER 2. INSAR : BACKGROUND 10
Figure 2.3: SAR interferogram imaging geometry in the plane normal to the flightdirection.
Figure 2.4 shows an example interferogram computed over the Cotton Bowl basin
in Death Valley (California) using two SAR images acquired using the SEASAT
satellite. The interferometric phase map shown in Figure 2.4 can be decomposed
into following components:
Δφ = φbase + φtopo + φdefo + φatm + φnoise (2.2)
where φbase is the phase difference due to the baseline between the positions of the
satellite corresponding to the two acquisitions, φtopo is due to topography, φdefo is due
to ground deformation, φatm is due to atmospheric delay, and φnoise is due to other
terms including the ionosphere and system noise.
For mapping topography, all terms besides φbase and φtopo are treated as noise
(Zebker and Goldstein, 1986).
φbase + φtopo = −4π
λB sin (θ − α) (2.3)
CHAPTER 2. INSAR : BACKGROUND 11
Figure 2.4: Example interferogram of the Cotton Bowl basin in Death Valley, CA(Goldstein et al., 1988). The salt flats in the center of the image show no fringeswhereas topographic fringes are clearly visible on the hills surrounding the basin.
where θ is the look angle and α is the angle that the baseline subtends with the
reference horizontal plane as shown in Figure 2.3 (Rosen et al., 2000). In fact, InSAR
was first developed to map the Earth’s topography (Zebker and Goldstein, 1986)
and was the technology behind the Shuttle Radar Topography Mission (SRTM)
(Farr et al., 2007). As can be seen in Figure 2.4, interferometric phase can only
be determined modulo 2π. To obtain a continuous interferometric phase map, the
differential phase between all neighboring pixels is integrated. This process is called
“phase unwrapping” and is discussed in detail in Chapter 5. The unwrapped phase
can then be converted to topography by inverting Equation 2.3.
For crustal deformation studies, φdefo is the only term of interest and other terms
are either corrected for or treated as noise. φbase can be estimated using precise orbit
information and an elliptical or spherical model for the Earth’s surface. The slightly
different imaging geometries also produce a slight parallax, if the area has topography.
A digital elevation model (DEM) such as from the Shuttle Radar Topography Mission
(SRTM) (Farr et al., 2007) or another interferogram (Gabriel et al., 1989) of the same
area can be used to estimate the φtopo term. This technique is called “Differential
CHAPTER 2. INSAR : BACKGROUND 12
Interferometry” or D-InSAR. This was first successfully applied to study the Landers
earthquake (Massonnet et al., 1993; Zebker et al., 1994).
2.4 Limitations of conventional InSAR
D-InSAR has been successfully applied for measuring ground deformation due to
active volcanism (Amelung et al., 2000; Pritchard and Simons, 2002), co-seismic
motions (Zebker et al., 1994; Simons et al., 2002), post-seismic motions (Pollitz et al.,
2001; Jacobs et al., 2002), mining and groundwater induced deformation (Amelung
et al., 1999), and creeping faults (Ryder and Burgmann, 2008; Burgmann et al., 2000).
The quality of our deformation estimates, measured by a quantity called “correlation”
(Zebker and Villasenor, 1992), also depends on the nature of the surface being imaged
itself. Change in the surface reflectivity with time or “temporal decorrelation”,
due to vegetation or melting of snow or other natural phenomena, decorrelates the
measurements significantly rendering the phase measurements unreliable. As a result,
most conventional InSAR studies tend to focus on dry and sparsely vegetated regions,
for example the southwest of the U.S., the Middle East and Tibet.
Degradation of the quality of interferometric phase or decorrelation also occurs
due to the variation in imaging geometry. Difference in incidence angles results in
the wavelets from the scatterers from a resolution element on the ground adding
up slightly differently and the measurements are not reproduced exactly. This
is called “spatial decorrelation” (Zebker and Villasenor, 1992) and increases with
increasing perpendicular component of baseline. A similar effect is also observed when
the imaging tracks for the two acquisitions are not completely parallel, producing
“rotational decorrelation”. A corresponding geometrical effect also occurs due to
a change in the squint angle, the angle with which the spacecraft points forward or
backward. The change in squint angle is characterized by a change in the SAR Doppler
frequencies leading to decorrelation. Although many of these decorrelation effects
can be minimized by filtering, there are critical limits on the baseline and Doppler
CHAPTER 2. INSAR : BACKGROUND 13
frequency differences beyond which no interferometric phase information can be
retrieved (Zebker and Villasenor, 1992). In short, the number of usable interferogram
pairs are limited by temporal and geometric decorrelation effects, effectively reducing
the temporal resolution of regularly acquired SAR data sets.
We denote the atmospheric phase “screen” (φatm) as the variation in the delay
of the signal as the signal propagates through the atmosphere, which introduces a
variable phase over the image (Hanssen, 2001). Most of the variation in the the
atmospheric phase term is due to the variation in the water vapor distribution over
the scene, and is often correlated with the topography of the area (Onn and Zebker,
2006). As the SAR acquisitions are separated by about 30 days, the atmospheric
phase screen at the two SAR acquisitions is essentially uncorrelated in time. A
common method of reducing the effects of the atmosphere is to combine information
from various interferograms using multi-temporal InSAR techniques, as described
in the next section. Errors in the known position of the imaging platform affect
the baseline estimates and typically manifest themselves as phase ramps in the final
interferograms. These phase ramps are a result of the wrong φbase used in phase
correction stage. Such ramps can be easily corrected by computing the fringe rate over
flat areas with no expected deformation and applying the correction over the entire
image. Residual errors from the phase ramp correction stage cannot be distinguished
from the atmospheric phase screen and multi-temporal InSAR techniques need to be
used to mitigate their effects.
2.5 Multi-temporal InSAR
Multi-temporal InSAR techniques are extensions of conventional InSAR aimed at
addressing the problems caused by decorrelation and atmospheric delay. These
techniques involve the simultanoues processing of multiple SAR acquisitions over the
same area to allow for the correction of uncorrelated phase noise terms and hence,
reduce errors associated with the deformation estimates. Currently, multi-temporal
CHAPTER 2. INSAR : BACKGROUND 14
Figure 2.5: Scattering mechanism models for a SAR resolution element - distributedscatterers (red), ideal single point scatterer (green) and persistent scatterer (blue).The persistently scattering pixel exhibits smaller phase variation than the distributedscattering pixel.
InSAR algorithms can be broadly classified into two categories - persistent scatterer
(PS) and small baseline (SBAS) methods. Each of these set of methods is designed
for a specific type of scattering mechanism.
The signal return from each resolution element on the ground depends on the
distribution and reflectivity of scattering centers within the element. Since we use
a monochromatic source, the reflected signal from each resolution element is the
coherent sum of individual wavelets scattered by numerous discrete scattering centers
within the element (Figure 2.5). Consider the effect of decorrelation by letting
scatterers in a resolution element move randomly with respect to the imaging platform
due to geometric and temporal decorrelation effects. If the resolution element were
composed of a single point scatterer (Figure 2.5), the received signal shows very little
variation with time. If this is the case, all interferograms can be used for estimating
deformation and we are able to estimate ground motion without error. But in reality,
such point scatterers rarely exist.
If all the scatterers are of comparable strength (Figure 2.5), then the interfero-
metric phase realizations are randomly distributed in the interval [−π, π). In such
cases, we can improve the signal-to-noise ratio (SNR) by averaging the interferometric
CHAPTER 2. INSAR : BACKGROUND 15
signal from adjacent resolution elements. Under this assumption random phase
contributions due to movement of scattering elements cancel out, leaving behind
the phase due to the average motion of the averaged resolution elements. This forms
the basis of short baseline (SBAS) methods. Stacking (Sandwell and Price, 1998)
is one of the simplest forms of SBAS methods. It determines an average velocity
model by averaging numerous interferograms with short orbital baselines to mitigate
atmospheric effects. Newer algorithms based on singular value decomposition and
temporal models (Berardino et al., 2002) have since been developed to estimate non-
linear deformation from a stack of interferograms.
On the other hand, if one of the scatterers in the resolution element is brighter than
the others (Figure 2.5), the interference from other scatterers is minimal, the received
signal is stable, and the motion of the dominant scatterer can be determined from the
interferometric phase. Such pixels form the model for Persistent Scatterer (PS) InSAR
(Ferretti et al., 2001). PS methods apply statistical techniques to identify pixels with a
single dominant scatterer that are relatively less affected by decorrelation, and use the
associated interferometric phase to infer a deformation time-series. For maximizing
our ability to identify such resolution elements, all interferograms are analyzed at the
highest possible resolution.
A detailed overview of the mathematical models used for PS-InSAR is provided in
Chapter 3. We also briefly describe popular PS-InSAR algorithms, most importantly
the Stanford Method for PS (Hooper, 2006) framework that forms the scaffold on
which rests the research described in this thesis.
Chapter 3
Mathematical modeling of persistent
scatterers
In this chapter, we describe in detail the mathematical models of scattering behavior
of persistently scattering pixels and derive the associated PS pixel amplitude and
interferometric phase statistics. We present models developed in earlier literature as
well as new ones developed during the course of this work. We introduce a physical
model for describing the nature of various components of observed interferometric
phase of a PS pixel. In the last section, we describe the salient features of the Stanford
Method for PS (StaMPS) framework (Hooper, 2006), which is used to estimate
the various phase components of the physical model. Our phase-stability based PS
selection techniques described in Chapter 4 builds on the StaMPS framework.
3.1 Introduction
Conventional InSAR (Section 2.3) is an effective method for measuring deformation
in areas of good coherence (Massonnet et al., 1993; Zebker et al., 1994). A closer
examination of any interferogram covering vegetated areas or areas prone to changes
in surface properties reveals the degradation of the quality of interferometric phase
due to changing surface properties with time, resulting in a loss of coherence (Zebker
and Villasenor, 1992). Variation in signal properties results from changes in the
16
CHAPTER 3. MATH MODELS 17
relative reflectivity of the scatterers and their relative positions within a resolution
element. If the scattered signal from a resolution element is dominated by a single
scatterer, the backscattered signal from the element is fairly consistent as described
in Section 2.5, and is less affected by spatial and temporal decorrelation. Hence, a
network of such single scatterer-dominated pixels may be used to extract deformation
signatures from interferograms that have been severley compromised by decorrelation.
The backscattered signal from each resolution element is represented by two
physical measures - the amplitude and the phase. We denote pixels that exhibit
statistical invariance in the observed SAR amplitude or interferometric phase over
a stack of SAR images as Persistent Scatterers (PS). A single point scatterer over
a perfectly dark background exhibits absolute invariance in the observed signal
characteristics due to differences in imaging geometry and with time. As the
background or the non-dominant scatterers in a resolution element get brighter, the
variance of the observed signal characteristics increases. We use mathematical models
described in this section to quantify this variation in signal properties and infer the
strength of the single dominant scatterer relative to the cumulative strength of the
other scatterers in the resolution element.
Observed interferometric phase measurements are affected by the imaging geome-
try, topography, atmospheric propagation and scatterer displacement or deformation.
In time-series InSAR, we deal with the problem of estimating a time-series of surface
deformation from a series of interferograms. A digital elevation model (DEM)
and precise orbit information are used to model and remove phase contributions
due to geometry and topography. Errors in the DEM and precise position of the
imaging platform also contribute to the observed interferometric phase. Hence, the
interferometric phase of a pixel (φint) in a differential interferogram can be represented
by (Hooper et al., 2004)
φint = φdef +Δφε + φatm +Δφorb + φn (3.1)
CHAPTER 3. MATH MODELS 18
where φdef represents the phase due to deformation, Δφε refers to the error introduced
by using imprecise topographic information, Δφorb refers to the error introduced due
to the use of imprecise orbits in mapping the contributions of Earth’s ellipsoidal
surface, φatm corresponds to the difference in atmospheric propagation times between
the two acquisition used to form the interferogram and φn represents the phase noise
due to the scattering background and other uncorrelated noise terms. The SAR
amplitudes are not affected significantly by the variations in imaging geometry, for
baseline values smaller than the critical baseline. Hence, their affects are neglected
when modeling SAR signals.
In persistent scatterer interferometry, our primary signal of interest is the return
from the dominant scatterer in the resolution element. The echoes from the dimmer
distributed scatterers in a pixel is also commonly referred to as “clutter” and
contributes to φn. Signal-to-clutter ratio (SCR) defined as the ratio between the
reflected energy from the dominant scatterer to that of the reflected energy from the
rest of the resolution element is a measure often used to indicate the strength of the
dominant scatterer in SAR pixels. High SCR (¿8) pixels exhibit low interferometric
phase variation (¡ 0.25 radians) and vice versa.
In Section 3.2 we describe the models used to describe the interferometric phase
components. In Section 3.3, we describe the scattering signal models that are useful
in analyzing both amplitude and interferometric phase of SAR pixels. The scattering
models in this section only addresses the clutter component (φn in Equation 3.1) of
the interferometric phase.
3.2 Interferometric phase components
In this section, we present phase models used to describe the first four phase
components of the estimated phase in differential interferograms (Equation 3.1)
(Hooper, 2006; Kampes, 2006). The first four phase components (Equation 3.1)
CHAPTER 3. MATH MODELS 19
are systematic and can be estimated accurately for a reasonably dense network of
medium signal-to-clutter ratio (SCR) pixels (SCR≥ 2).
3.2.1 Deformation
Deformation (φdefo) is the primary phase component or signal of interest in this work.
Persistent scatterer InSAR permits estimates of deformation time-series for a region
of interest by combining information from multiple SAR acquistions (Ferretti et al.,
2001). PS InSAR is an ideal tool for studying the spatial-temporal characteristics of
crustal deformation a region of interest, a few square km in size.
Temporal models for deformation like a constant velocity model with or without a
sinusoidal component for seasonal ground water fluctuations are often used to simplify
the estimation of the relevant deformation signals (Ferretti et al., 2001; Kampes,
2006; Colesanti et al., 2003; Werner et al., 2003). A priori information related to the
region of interest can also be used to develop structural models for deformation signal
estimation (Shanker and Zebker, 2010).
In this work, we use the StaMPS framework (Hooper, 2006) which provides a
means of estimating the deformation of a region of interest by assuming that deforma-
tion has a long wavelength spatial and low frequency temporal structure (Section 3.4).
Phase contributions due to the scatterer geometry errors (Section 3.2.2), atmospheric
propagation (Section 3.2.3) and errors in the precise orbits (Section 3.2.4) affect the
deformation phase estimates from a single interferogram (Hanssen, 2001). PS-InSAR
uses information from multiple interferograms to separate out the deformation phase
signal from these other phase contributions.
3.2.2 Geometric phase component (Δφε)
The phase contributions due to the imaging geometry and topography are removed for
interferograms used in time-series analysis by a process called flattening (Section 2.3).
This correction is applied in two stages. In the first stage, the phase component due
to the Earth’s curvature (φbase in Equation 2.2) is removed by assuming that all the
CHAPTER 3. MATH MODELS 20
scatterers lie on the WGS-84 ellipsoid. In the next stage, a digital elevation model
(DEM) of the area in the radar coordinates is used to compute the corresponding
phase component (φtopo in Equation 2.2). This second processing stage contains two
types of topography-related errors in the processed differential interferograms: DEM
error and sub-pixel position error.
Figure 3.1: Example inversion of observed interferometric phase (black dots) forgeometric phase components (solid line) as a function of the perpendicular baseline.
DEM error
Using an approximate DEM can introduce systematic phase artifacts. In many active
regions of the planet, the topography has significantly changed by a few meters from
the time when the digital models were created, for example using SRTM data (Farr
et al., 2007). If Δh is the error in the DEM used, the corresponding phase error
(Kampes, 2006) is
Δφε =4π
λB⊥ (θ)
Δh sin (θ)
r(3.2)
where B⊥ represents the perpendicular baseline, r represents the absolute one way
range from the satellite to the pixel under consideration and θ represents the look
angle.
All the data sets discussed in this thesis were acquired using the ERS and EnviSAT
satellites. The baselines of both the European Space Agency (ESA) satellites are well
controlled and do not exhibit a systematic drift in baselines. This is not the case for
CHAPTER 3. MATH MODELS 21
most SAR systems, including the ALOS PALSAR system, which has a characteristic
baseline drift. If the perpendicular baselines are correlated with time and the area
being analyzed is characterized by steady deformation, the effects of the deformation
phase component and the geometric error components are indistinguishable.
Sub-pixel range position error
Range resolution (Δr) is one of the important performance characteristics of a SAR
system and limits the accurate positioning of a dominant scatterer in a SAR image.
The location of the dominant scatterer in a radar pixel may not always correspond
to the center of the mapped pixel location on the ground. Hence, if the dominant
scatterer within a SAR pixel is located at a distance ε from the center of the pixel,
the corresponding systematic phase offset (Kampes, 2006) is given by
Δφε =4π
λB⊥ (θ)
ε cos (θ)
r(3.3)
Since both the phase terms 3.2 and 3.3 have the same functional relationship with
the perpendicular baseline and the look angle, they are indistinguishable. For most
deformation monitoring applications, the additional processing step of distinguishing
between the height and the sub-pixel position components is not necessary. The
combined phase contributions can be estimated using a non-linear estimator as shown
in Figure 3.1 (Hooper, 2009). It is not possible to use a simple linear estimator due
to the possible wrap around of phase as a result of large perpendicular baseline values
or DEM errors.
Sub-pixel azimuth position error
An error term similar to the DEM error and sub pixel range position arises if
the acquisition squint angles are different for the scenes combined to form an
interferogram and the dominant scatterer is not located at the center of the SAR
pixel. The sub-pixel azimuth position error only produces a systematic phase offset
CHAPTER 3. MATH MODELS 22
if the SAR scenes are processed to a geometry that is different from the acquisition
geometry. For time-series InSAR, the SAR scenes are often processed to the zero-
Doppler geometry for reducing data volume and hence, only the range difference
to the center of the SAR pixels are used for phase compensation. If the dominant
scatterer is offset from the pixel center by a distance of η in the azimuth direction,
the corresponding systematic phase shift is given by
Δφν =2π
v·ΔFDC · η (3.4)
where ΔFDC corresponds to the difference in the Doppler centroid frequencies of the
slave and the master scenes, and v corresponds to the velocity of the SAR platform
(Kampes, 2006). The data sets discussed in this thesis were acquired using the
ERS and Envisat satellites, which are characterized by slow drift in the Doppler
centroid. Hence, the phase component is correlated with the time-difference between
acquisitions and can be mistaken for time-dependent deformation. These phase errors
are much smaller in magnitude (order of 0.2 radians) than deformation and hence,
we treat it as noise. We do not attempt to correct for these systematic offsets. An
oversampling and local maxima detection approach (Ketelaar, 2009) can be used to
determine the sub-pixel position of the dominant scatterer and the corresponding
phase corrections can be applied if needed.
3.2.3 Atmospheric phase screen
The dominant error source in many interferograms results from the spatial heterogene-
ity of the wet component of atmospheric refractivity, resulting in excess path length
of the radar signal propagating through the neutral atmosphere (Goldstein, 1995;
Onn and Zebker, 2006). The atmospheric phase signal varies gradually over space
and is often modelled as a long wavelength component in unwrapped phase (Hooper,
2006; Onn and Zebker, 2006). Emardson et al. (2003) estimated the correlation
function of the atmospheric phase screen using InSAR and GPS data, and for data
CHAPTER 3. MATH MODELS 23
over Southern California estimated a phase difference of approximately one radian at
1 GHz for points separated by a distance of two km.
It has been shown that the atmospheric propagation delay between two points
increases exponentially with the distance separating them, almost linearly with their
altitude difference and is independent of the wavelength (Emardson et al., 2003; Onn
and Zebker, 2006). We ignore the systematic variation of the tropospheric phase delay
due to topography in most cases and treat it as spatially correlated noise (Hooper,
2006) unless specifically mentioned otherwise. This is a reasonable assumption as
the topgraphy in most areas that we analyze in this work do not vary by more than
500 m over the area of interest. In regions, where the topography varies significantly
over the scene of interest, we use a model in which the tropospheric phase delay is
linearly proportional to the altitude of the SAR pixel. More details of this correction
technique will be presented below as a part of the discussion of results corresponding
to landslides in Norway (Chapter 6).
3.2.4 Orbital errors
The positions of the SAR platform must be known with sub-meter accuracy for precise
determination of baselines and accurate topographic phase correction (Rosen et al.,
2000). Geographically correlated orbit errors, like those due to gravity models cancel
each other out in baseline computations (Hanssen, 2001). However, non-conservative
forces like time-variable drag and time-variable solar radiation pressure cannot be
modelled accurately in the current precise orbit determining systems (Scharroo and
Visser, 1998). This error in the precise positions of satellites also propagates into
the baseline calculations, and hence affects the topographic phase correction and the
Earth ellipsoid correction.
Along-track position error
Along-track positioning errors can be corrected in the coregistration process, where
the absolute shift between the two SAR scenes in the azimuth direction are also
CHAPTER 3. MATH MODELS 24
estimated. The across-track error of the master scene affects the DEM (in radar
coordinates) that is used to generate the differential interferogram. However, these
errors are in the order 1-2 m, which is significantly smaller than the SRTM DEM
spacing of 30 m (Farr et al., 2007). Hence, the phase error due to the approximate
DEM is expected to be small. This DEM error (Equation 3.1) can be estimated as
they are correlated with the perpendicular baseline values. The across-track orbit
errors produce systematic errors and can be estimated from the data itself.
Across-track position error
The interferometric phase is much more sensitive to across-track errors than along-
track errors and particularly, those in the vertical direction. These errors manifest
as a phase ramp superimposed over the actual interferometric phase, as the incorrect
baseline is used for correcting the contribution of the spherical or ellipsoidal earth
(Δφbase in Equation 2.2). These errors introduce a fairly gentle phase ramp when
precise orbits are used with ERS or Envisat data (Scharroo and Visser, 1998). This
signal is indistinguishable from phase gradients introduced by the atmospheric phase
delays or long wavelength deformation (Hanssen, 2001). When analyzing areas
of 40 km × 40 km in size, as is typical in this work, we do not encounter any
large phase ramps and treat this phase component as noise that is indistinguishable
from the atmospheric component. Occasionally, large phase ramps are encountered
which can be corrected by fitting a plane or a parabolic surface to the unwrapped
interferogram (Hanssen, 2001) or by explicitly counting fringes in the wrapped
inteferogram (Kohlhase et al., 2003). However this method can be severely limited by
presence of large decorrelated areas (Kohlhase et al., 2003) and in vegetated terrain.
3.2.5 Other phase terms
We have, so far, described all the systematic phase components that affect the
observed differential interferometric phase (Equation 3.1). There are numerous other
phase aberrations introduced by the SAR and InSAR processing implementations
CHAPTER 3. MATH MODELS 25
(Bamler and Just, 1993). Various effects such as geometric misregistration, defocusing
in azimuth and errors in range migration can introduce phase aberrations. With
improved SAR processing algorithms and improved precise orbit determination, many
of these artifacts are effectively reduced. Yet in combination with thermal noise, these
factors contribute to an interferometric phase noise “floor”. These effects are less
apparent in conventional multi-looked interferograms where the signal-to-noise ratio
(SNR) is boosted by averaging pixels. These effects are more visible when handling
single-look interferograms (see Chapter 7). These small phase effects are treated
similar to the scatterer noise (φn), as they are assumed to be uncorrelated with time.
3.3 Scattering Signal Models
The phase noise (φn) term in Equation 3.1 can be primarily attributed to non-
dominant scatterers in the resolution element. Published PS-InSAR algorithms
generally represent this phase component using one of two PS signal models - the
constant signal model and the Gaussian signal model. In this section, we discuss
the probability distribution functions (PDFs) for the distribution of the SAR pixel
amplitude and scatterer phase noise for both of these signal models. PS pixels can be
defined as those resolution elements in a series of SAR images that are less affected
by scatterer noise and other noise factors.
These signal models represent the signal return from the resolution elements.
These signals are also affected by atmospheric propagation and other physical noise
sources before reaching the SAR antenna. For simplicity, we can group all temporally
and spatially uncorrelated noise sources with the scatterer noise.
3.3.1 Constant Signal Model
This signal model for persistent scatterers was first suggested by Ferretti et al. (2000)
and Ferretti et al. (2001). Let z = 1+ n represent the return from a single resolution
element in a single look complex SAR image. The radar return from the dominant
CHAPTER 3. MATH MODELS 26
scatterer is assumed to be a constant, over a small range of imaging geometries and
baseline, and is normalized to unity. “n” represents the noise from the other scatterers
and uncorrelated noise sources. The noise term is assumed to be circular Gaussian
(complex number with the real and imaginary parts represented by i.i.d zero mean
Gaussian variables) with variance σ2n. The SCR of this signal model is given by 1/σ2
n.
SAR amplitude
The amplitude of the signal (b = ||z||) is given by the Rice distribution (Papoulis,
1991) and can be defined using the modified Bessel function (I0) as follows:
fb (b) =2b
σ2n
· I0(2b
σ2n
)· e−(1+b2)/σ2
n (3.5)
SAR pixel phase
The corresponding SAR pixel phase ( � z = θ) distribution can be written in terms of
error functions (Abramowitz and Stegun, 1972) :
fθ (θ) =1
2πexp
(−sin2 θ
σ2n
)·[exp
(− cos2 θ
σ2n
)+
√π
σ2n
· cos θ · erfc(−√
1
σ2n
· cos θ)]
(3.6)
InSAR pixel phase
The intereferometric phase distribution corresponding to this signal model does not
have a simple closed form expression. A derivation of a semi-analytical solution for
this PDF is presented from first principles in this work. If z1 = r1 ·ejθ1 and z2 = r2 ·ejθ2represent the signal return from two different SAR images for the same resolution
element or pixel, then the distribution of the interferometric phase (φn = � (z1 · z∗2))can be derived starting from the SAR pixel phase (Equation 3.6) as shown below. As
the phase observations in the individual SAR images are independent, we write joint
distribution of two independent SAR phase observations θ1 and θ2 as
fθ1,θ2 (θ1, θ2) = fθ (θ1) · fθ (θ2) (3.7)
CHAPTER 3. MATH MODELS 27
φn and φsum, a new random variable, can be defined in terms of θ1 and θ2 as
φn = (θ1 − θ2)2π
φsum = (θ1 + θ2)2π (3.8)
We now apply this transformation to obtain the joint PDF of φn and φsum. The
modulo 2π operation, represented by (·)2π, ensures that φn and φsum are restricted to
[−π, π). The tranformation uses the Jacobian of Equation 3.8 (Papoulis, 1991) and
is given by
fφn,φsum(φn, φsum) =1
2
⎡⎣ fθ1,θ2
(φn+φsum
2, φsum−φn
2
)+fθ1,θ2
(φn+φsum
2+ π, φsum−φn
2+ π
)⎤⎦ (3.9)
Equation 3.9 can be numerically integrated over φsum to yield the PDF of the the
phase difference φn as follows:
fφn(φn) =
∫ π
−π
fφn,φsum (φn, φsum) · dφsum (3.10)
These probability distribution functions can be pre-computed for various noise levels
(σ2n) and stored for later applications. PDFs of the SAR amplitude and interferometric
phase for various values of SCR (γ) are shown in Figure 3.2.
3.3.2 Gaussian signal model
Let z = s + n represent the return from a single resolution element in a single look
complex SAR image. In this case, the return from the dominant scatterer is also
modeled by a circular Gaussian random variable, with variance σ2s . This assumption
was shown to be valid for SAR data by Sarabandi (1992).
SAR amplitude
The amplitude of the signal is characterized by the Rayleigh distribution (Abramowitz
CHAPTER 3. MATH MODELS 28
Figure 3.2: (Top) PDFs of the SAR amplitude function and (Bottom) PDFs of theinterferometric phase residuals for the constant signal model evaluated for SCR valuesof 0.5, 4 and 100.
and Stegun, 1972) as follows:
fb (b) =2b
σ2s + σ2
n
· exp( −b2
σ2s + σ2
n
)(3.11)
Since z is also distributed as a circularly Gaussian variable, the phase of the SAR
pixel is distributed as a uniformly random variable over [−π, π).
Interferometric phase
The PDF of the inteferometric phase of a single look pixel in this model is given by
(Just and Bamler, 1994; Lee et al., 1994)
fφ (φ) =1− |ρ|2
2π· 1
1− β2φ
·⎡⎣1 + βφ · arccos (−βφ)√
1− β2φ
⎤⎦ (3.12)
where βφ = |ρ| · cosφ
CHAPTER 3. MATH MODELS 29
The magnitude of the interferometric correlation (|ρ|) depends on the signal to clutter
ratio (SCR or γ) following Just and Bamler (1994) and Lee et al. (1994) :
γ =|ρ|
1− |ρ| (3.13)
PDFs of the SAR amplitude and interferometric phase for various values of SCR are
shown in Figure 3.3. Equation 3.12 reduces to a Dirac delta function for perfectly
correlated signals( |ρ| = 1) and to a uniform distribution on [−π, π) for perfectly
decorrelated signals (|ρ| = 0).
Figure 3.3: (Top) PDFs of the SAR amplitude function and (Bottom) PDFs of theinterferometric phase residuals for the constant signal model evaluated for SCR valuesof 0.5, 4 and 100.
CHAPTER 3. MATH MODELS 30
3.4 StaMPS framework
The Stanford Method for PS (StaMPS) framework was initially developed for PS
applications in natural terrain (Hooper et al., 2004; Hooper et al., 2007) and since,
has been expanded to include short baseline analysis (Hooper, 2008). For PS-InSAR
analysis, a set of N common-master interferograms, where the master scene has been
chosen to minimize the decorrelation effects (Hooper et al., 2007), is analyzed at the
highest possible resolution. We generalize Equation 3.1 to
φx,i = φdefo,x,i +Δφε,x,i + φatm,x,i +Δφorb,x,i + φn,x,i (3.14)
where x represents the spatial index of the pixel under consideration in interferogram
i and all the other symbols are the same as given in Equation 3.1. PS pixels are
again defined as pixels with very little variation in the scatterer noise term (φn,x,i).
The first four terms in Equation 3.14 dominate the scatterer term and need to
be reliably estimated for identification of PS pixels. The StaMPS framework is a
collection of spatial and temporal filtering routines that allow us to estimate each of
these phase components by assuming a spectral structure. Table 3.1 describes the
spectral characteristics assumed for each of the phase components. All the observed
interferometric phases are wrapped values and hence, the low pass components from
Table 3.1 are estimated using a combination of a low pass filter and an adaptive phase
filter (Hooper et al., 2007) that preserves the interferometric fringes.
Table 3.1: Spectral characteristics for various phase components of the observedinterferometric phase for a PS pixel (Hooper, 2006).
Component Spatial Properties Temporal Propertiesφdefo,x,i Deformation Low freq Low freqφatm,x,i Atmosphere Low freq High freqΔφorb,x,i Orbital errors Low freq High freqφn,x,i Scatterer noise High freq High freqφε,x,i DEM Error High freq Correlated with baseline
CHAPTER 3. MATH MODELS 31
From Table 3.1, it is clear that the deformation signal, atmospheric phase screen
and the orbital errors are all spatially correlated and can be estimated simultaneously
as a combined term for each interferogram. The DEM error term can be estimated
on a pixel-by-pixel basis. The various steps involved in the estimation of the spatially
correlated phase terms are shown in Figure 3.4 and described in the following
subsections.
3.4.1 Preliminary PS candidate selection
Amplitude dispersion, defined as the ratio between the standard deviation and the
mean of the SAR amplitude (Ferretti et al., 2001), is computed on a pixel-by-pixel
basis and a preliminary set of candidates are identified with a threshold of typically
0.4 to 0.45. This is a very liberal threshold compared to that typically suggested
for PS selection (Ferretti et al., 2001) and primarily eliminates areas over water and
heavily decorrelated pixels in vegetated areas. Accurate computation of amplitude
dispersion requires precise amplitude calibration of the SAR images using the antenna
gain pattern. Since we use a relaxed threshold for candidate selection,we can use a
simplified constant calibration factor obtained by averaging the pixel brightness in
the SAR images (Lyons and Sandwell, 2003). This candidate selection stage is purely
optional, but often decreases the processing time and memory requirements by a
factor of ten.
3.4.2 Weighted estimation of correlated phase terms and
DEM error
The first three terms in Equation 3.14 are spatially correlated over short distances and
can be estimated by spatial filtering of each interferogram. Lyons and Sandwell (2003)
showed that the noise in interferometric phase observations can be significantly
reduced by weighting the pixel amplitudes by a measure of their stability, before
multilooking or adaptive filtering in the complex image domain. Building on this idea,
CHAPTER 3. MATH MODELS 32
the amplitudes of the pixels are weighted by the inverse of the amplitude dispersion
(Hooper, 2009). Once the spatially correlated terms are estimated and subtracted,
the baseline diversity of the stack of interferograms is exploited to determine the
phase terms associated with the DEM error (Δφε) for each selected candidate pixel
in the interferograms. The phase residuals are interpreted as scatterer noise terms
φn for each pixel and used to compute a temporal coherence (ρtemp) measure on a
pixel-by-pixel basis as follows:
ρtemp,i =N∑k=1
[ejφn,k,i
N
](3.15)
We estimate the probability of estimating a similar coherence value for a purely
random phase sequence by numerical simulation of a 100, 000 random phased
sequences. Hooper et al. (2007) empirically observed that the spatially correlated
phase estimation is significantly improved and converged faster if the amplitude
weights are readjusted using this probability measure. This process is repeated
iteratively until the estimated probability values converge (Figure 3.4). For more
details of the iterative estimation of correlated phase terms, we refer the readers to
Hooper (2006) and Hooper et al. (2007).
3.4.3 PS selection
The StaMPS framework is self-sufficient and provides a method for identifying the
PS pixels after estimation of the spatially correlated phase terms. However, the most
important contribution of the StaMPS method is the estimation of spatially-correlated
phase terms as described in the previous section. Figure 3.4 summarizes the various
steps involved in estimating the spatially correlated phase terms. In this work, we use
our own PS selection algorithms but depend on the ability of the StaMPS framework
to reliably estimate spatially correlated phase terms. A detailed discussion of the PS
selection techniques is provided in Chapter 4.
CHAPTER 3. MATH MODELS 33
Figure 3.4: The StaMPS framework for estimating spatially correlated phase termsfor every PS candiate in each interferogram. The outputs of this processing stepinclude estimates of spatially correlated terms, geometric errors and pixel coherences.
3.4.4 Sidelobe effects
The impulse function of a single target is characterized by sinc functions in the range
and azimuth directions. Hence, any SAR image can be represented as a convolution
of the reflectivity of the scatterers on the ground and the two dimensional sinc
function (Cumming and Wong, 2005). As a result, bright scatterers in the ground
dominate the response from the surrounding darker scatterers. To avoid the selection
of pixels affected by sidelobes and misinterpretation of systematic phase artifacts as
deformation, whenever adjacent pixels are identified as PS, the StaMPS framework
CHAPTER 3. MATH MODELS 34
discards the less coherent of these pixels as affected by sidelobes of the brighter and
more coherent pixels.
The wrapped phase values corresponding to the final reliable network of PS
are then unwrapped and smoothened using appropriate spatial and temporal filters
to estimate the line of sight deformation from the stack of interferograms. Phase
unwrapping is the subject of a later chapter in this work (Chapter 5).
3.4.5 Other PS frameworks
Many other PS-InSAR algorithms and frameworks have been developed and suc-
cessfully applied to study temporal characteristics of deformation across the world.
The more popular of these include the original Permanent ScatterersTMfrom Tele-
Rilevamento Europe (TRE), the Interferometric Point Target Analysis (IPTA)
framework (Werner et al., 2003) from GAMMA software, the GENESIS-PSI system
(Adam et al., 2003) from the German Aerospace Center (DLR) and the DePSI system
from Delft University of Technology (Kampes, 2006). All these PS systems are
designed to identify pixels with stable amplitude characteristics and use amplitude
statistics to identify coherent scatterers. StaMPS (Hooper, 2006) was the first
framework to consider phase stability as a criterion for coherent scatterer selection
and incorporate information from neighboring pixels to determine phase stability of
each pixel. The full resolution SBAS (Lanari et al., 2004b) is another full resolution
time-series InSAR technique but uses all small geometric baseline interferograms to
reduce the effects of decorrelation. As this system does not operate on single-master
interferograms, it is technically not a PS system. More details on the PS selection
mechanisms of these frameworks will be discussed in Chapter 4.
3.5 Summary
We have presented here two mathematical models – constant signal model and
Gaussian signal model, to describe the scattering properties of persistently scattering
CHAPTER 3. MATH MODELS 35
pixels. The distribution functions for the SAR amplitudes and interferometric phases
for both the signal models have been described. The PDF of the interferometric phase
for the constant signal model has been presented for the first time in literature, in
this work. These signal models can be used as a reference model for quantifying the
relative strength of the dominant scatterer in pixels and for quantifying the variation
in signal amplitude and phase. Each signal model represents a slightly different
interpretation of the received radar echo, but the PS selection experiments described
in Chapter 4 do not indicate that either of the signal models is superior to the other.
The two models are almost equivalent with the Gaussian signal model exhibiting
roughly twice the phase variation as the constant signal model for the same SCR.
We have also described the salient features of the StaMPS framework (Hooper,
2006) in this chapter. The StaMPS framework is used to model the effects of imaging
geometry, atmospheric propagation and orbital errors on the observed interferometric
phase in terms of their spectral structure without assuming a fixed functional form.
Chapter 4
Persistent Scatterer Selection
In this chapter we present a new information theoretic approach to PS selection.
Building on the mathematical models presented in Chapter 3, we present the most
common amplitude-based PS selection methods and then describe our new phase-
based approach for identifying PS pixels in a stack of common master single-look
inteferograms by comparing the observed interferometric phase against a known
mathematical model. Our method identifies a denser PS network in natural terrain
than other published algorithms (Shanker and Zebker, 2007).
4.1 Amplitude based PS Selection
In this section, we provide a brief overview of the PS selection methods suggested
in literature so far. Most of these PS selection algorithms are amplitude-based and
operate on coregistered SAR images.
4.1.1 Amplitude Dispersion
Amplitude dispersion was the first method developed to identify PS pixels in a series
of SAR images (Ferretti et al., 2000). The constant signal model (Section 3.3.1) is
used to characterize the statistical behavior of PS pixels. For low SCR pixels, the
amplitude distribution (Equation 3.5) tends to a Rayleigh distribution (Papoulis,
1991) that depends only on the noise variance (σ2n). For high SCR pixels, as is
36
CHAPTER 4. PERSISTENT SCATTERER SELECTION 37
Figure 4.1: Amplitude dispersion (DA) and phase dispersion (σφ) as a function ofincreasing noise for the constant signal model (Section 3.3.1) for a simulated stack of25 interferograms. For low noise values, the two curves track each other well.
often the case in urban regions, the amplitude distribution approaches a Gaussian
distribution (Ferretti et al., 2001) and the phase deviation (σφ) can be approximated
by
σφ ≈ σn =σb
μb
= DA (4.1)
where μb and σb represent the mean and standard deviation of the SAR pixel
amplitude and DA represents the amplitude dispersion. Figure 4.1 shows the
relationship between phase deviation (σφ) and amplitude dispersion (DA). From
this image, it is clear that the one-to-one correspondence between the phase statistics
and the amplitude statistics breaks down beyond a dispersion value of 0.25 or a
corresponding signal-to-clutter ratio (SCR) value of 8. This selection criterion works
well in urban areas where man-made structures have higher reflectivity and hence,
high SCR. In natural terrain, there could potentially exist low SCR PS pixels with
stable phase characteristics which would not be identified by this method.
Precise radiometric calibration of the coregistered SAR images is essential for
reliable estimation of pixel amplitude statistics and the identification of persitent
scatterers in the area of interest. In areas with calibrated corner reflectors or
CHAPTER 4. PERSISTENT SCATTERER SELECTION 38
transponders, external calibration (Freeman, 1992) can be applied and the scaling
constants for each SAR scene can be obtained by the analysis of uncalibrated SAR
scenes themselves (Freeman, 1992). This scaling constant can be derived using the
peak power or the integral output of the reference targets (Ulander, 1991). But
often, this method cannot be applied for calibrating SAR images due to the absence
of calibrated targets in the area of interest.
To overcome this calibration issue, the PS selection is implemented iteratively. If
the truly bright targets (PS) correspond to the constant signal model 3.3.1, then their
amplitude distribution is given by the Rice distribution. Hence, for each iteration a
set of calibration scaling constants can be identified in the least squares sense, such
that the observed amplitude characteristics are best fit by a Rice distribution (D’aria
et al., 2009). The availability of a large number of SAR scenes can thus be exploited
to derive the SAR calibration constants as well. The amplitude dispersion method
has also been adapted for other PS frameworks (Kampes, 2006; Crosetto et al., 2003;
Lyons and Sandwell, 2003).
4.1.2 Signal-to-Clutter Ratio
This set of PS selection techniques is also inspired by SAR calibration techniques. The
SCR approach has been used by researchers at DLR in their GENESIS processing
chain (Adam et al., 2003) and at Delft University in the DePSI processing chain
(Kampes, 2006). The SAR images are oversampled by a factor of 2 to reduce aliasing
effects due to the complex multiplication of the SAR images (Nutricato et al., 2002).
The signal-to-clutter ratio (SCR) is estimated by computing the ratio of the power
of a PS candidate with that of its immediate neighboring pixels (Adam et al., 2003;
Kampes and Adam, 2005). As a result of the oversampling, the sub-pixel position of
the dominant scatterer (Ketelaar, 2009) is also estimated during the processing. A
typical SCR threshold of 2.0 (Kampes, 2006; Ketelaar, 2009) is used to identify the
PS pixels from amongst the candidates. The main advantage of this method is that
CHAPTER 4. PERSISTENT SCATTERER SELECTION 39
Figure 4.2: The Signal-to-Clutter Ratio for the pixel of interest (black) is computed byratio of the energy in the sidelobes (white) and the background (gray) in oversampledSAR images.
the SCR determination is independent of the calibration constant as the brightness
of each pixel is compared to those of its immediate neighbors.
Implementation of this method in heavily urbanized areas, however, is difficult due
to the presence of bright scatterers in close proximity to each other. Sophosticated
SCR estimators and filters are needed in such scenarios for effective estimation of
PS pixels. Kampes (2006) provides a comparison of the SCR technique against the
amplitude dispersion technique.
4.1.3 Scripps technique
The Scripps PS technique (Lyons and Sandwell, 2003) is a not a conventional PS-
InSAR technique by definition. It is a filtering framework that uses the amplitude
stability characteristics of each pixel to reduce noise in the multi-looked phase of
differential interferograms. The cleaner differential interferograms are then used for
unwrapping and stacking to determine the average line of sight (LOS) deformation
velocity.
CHAPTER 4. PERSISTENT SCATTERER SELECTION 40
The amplitude of the coregistered SAR images are roughly calibrated by
normalizing them with the sum of the brightness values of all pixels in the scene
of interest. A scattering function (s), defined as the inverse of ampitude dispersion
(Section 4.1.1), is computed for each pixel over this set of calibrated coregistered
images. The single-look high resolution interferograms are multilooked after scaling
the amplitude by s2 for each of the pixels. These multi-looked interferograms are then
unwrapped and averaged to estimate an average velocity. This averaging step can
also be interpreted as a simple time-series InSAR technique where a linear temporal
deformation model is used. The main difference with the conventional PS-InSAR
techniques however, is that this framework allows for use of multi-master multi-looked
interferograms. The weighted multi-looking scheme introduced by this technique was
the predecessor on which the StaMPS filtering framework described in Section 3.4
was built.
4.2 Elimination using a temporal model
Very often, the criteria mentioned in the previous section are used with liberal
thresholds to select a dense network of PS pixel candidates. If reliably unwrapped
interferograms are available, all the pixels with unwrapped phase values that do not
reasonably fit a pre-determined deformation temporal model can be rejected as non-
PS. In the absence of unwrapped interferograms, the time-series of the differences
in interferometric phase for pixels in geographical proximity (also called double-
differences) are analyzed and a temporal deformation model (typically linear) is
used to approximate the interferometric phase (Werner et al., 2003; Kampes and
Adam, 2005). Double-difference time-series typically do not involve multiple cycle
phase jumps and allows us to work with wrapped data. Mathematically, the method
involves the analysis of the spatial gradient of the differential interferometric phase
(Equation 3.1)
∇φifg = ∇φdef +∇ (Δφε) +∇φatm +∇ (Δφorb) + n (4.2)
CHAPTER 4. PERSISTENT SCATTERER SELECTION 41
The model assumes that the phase gradients of the atmospheric phase screen (∇φatm)
and the orbit errors (∇ (Δφorb)) for nearby pixels are negligible compared to the
other phase components. Hence, the phase gradient term can be approximated by a
temporal deformation model and a linear DEM error term. It is to be noted that the
temporal model regards the spatial gradient as a function of time and not the actual
deformation itself. This model is applied to all the pixel pairs that are separated by a
pre-determined finite distance. The reliability of the model for a corresponding phase
can then be quantified by a numerical measure called temporal coherence defined as
ρtemp = ‖Nifg∑k=1
e[φifg,k−φmodel,k]‖ (4.3)
where is φifg,k represents the observed double-difference phase in interferogram k and
φmodel,k represents the modeled phase for the same pair of pixels in interferogram k.
The temporal coherence value ranges from zero to one and represents the quality of
fit of the model to the observed data. Amongst the initial set of PS candidate pairs,
many exhibit large residuals or a low temporal coherence. Pixels that figure in a
large number of such low coherence pairs are flagged as noisy and not included in the
final PS network. Thus, a deformation model incorporating any a priori information
about the area being analyzed itself can also be used for identifying a PS network.
4.3 StaMPS PS selection
The StaMPS framework, as developed by Hooper (2006) is described in detail in
Section 3.4. The StaMPS method is a complete package of PS-InSAR analysis tools
and incorporates vital aspects of many other PS selection algorithms. StaMPS was
the first method to adopt phase stability and spatial correlation as a criterion for PS
selection.
Equation 3.15 defines the pixel-wise coherence, a phase stability measure ranging
between zero and one corresponding to a purely random phased pixel and an ideal
CHAPTER 4. PERSISTENT SCATTERER SELECTION 42
point scatterer respectively. Similar coherence measures are also computed for a few
hundered thousand randomly simulated phase sequences. Due to the finite number
of phase observations, i.e, the number of interferograms, the observed estimates are
biased (Touzi et al., 1990). As a result, it is assumed that all the coherence estimates
with values less than 0.3 corresponds to noisy non-PS pixels. This assumption is
used to normalize the histograms of the observed coherence values and the simulated
random phase coherence values. Thus the probability of each pixel belonging to the
family of PS pixels, based solely on the coherence measure, can be determined as
shown below (Hooper, 2006).
P (x ∈ PS| ρ) = 1−[ ∫ 0.3
0Pobs (ρ) · dρ∫ 0.3
0Psim (ρ) · dρ
]· Psim (ρ)
Pobs (ρ)(4.4)
Psim and Pobs correspond to the simulated coherence histogram and the observed
coherence histogram respectively. Based on these probability measures, we can define
a random pixel acceptance rate (RPAR) corresponding to the coherence threshold
(ρthr) as
RPAR (ρthr) =
[ ∫ 0.3
0Pobs (ρ) · dρ∫ 0.3
0Psim (ρ) · dρ
]·[ ∫ 1
ρthrPobs (ρ) · dρ∫ 1
ρthrPsim (ρ) · dρ
](4.5)
The RPAR values can be numerically calculated using the observed coherence values
and an appropriate threshold for PS selection can be chosen depending on a suitable
choice of RPAR. Hooper (2006) also observed that the selection process can be
improved by incorporating the amplitude information. All pixels with coherence
values greater than the estimated threshold (ρthr) are labelled as PS pixels.
The StaMPS PS selection technique outperforms the other PS algorithms
described in the previous sections and successfully identifies more PS pixels in
non-urban vegetated terrain. We build on the StaMPS framework laid out by
Hooper (2006) and define our own PS selection algorithm. The details of our selection
algorithm are presented in the next section.
CHAPTER 4. PERSISTENT SCATTERER SELECTION 43
4.4 Maximum Likelihood PS selection
Once all the spatially correlated terms are estimated as described in Equation 3.4, we
are left with only the noise term, φn, representing the combined phase contribution
from the scatterers in the resolution cell. We compare the observed phases with the
theoretical phase distributions (Equations 3.10 and 3.12) to obtain the maximum
likelihood estimate of SCR (γ).
From Equation 3.10 and Equation 3.12, we have a forward model for the
distribution of interferometric phase as a function of γ, P (φn|γ). Using Bayesian
concepts, we estimate the value of γ maximizing the P (γ|φn1 , · · · , φnN), where
φn1 , · · · , φnNare the residual noise phase terms at a pixel in N interferograms, γ
is the SCR of the brightest scatterer in the resolution element, and P (A|B) is the
probability of event A given event B. In other words, we determine the most likely γ
that produced the observed distribution of residual noise phase terms. Using Bayes’
rule we rewrite the probability function as follows:
P (γ|φn1 , · · · , φnN) =
P (φn1 , · · · , φnN|γ) · P (γ)
P (φn1 , · · · , φnN)
(4.6)
The term in the denominator of the right hand side of Equation 4.6 is independent
of γ, thus we need only to maximize the numerator over all values of γ. As we have
no prior knowledge of γ, we assume that all values are equally likely, that is, P (γ) is
constant for all values of γ. We also assume that the observed interferometric phase
values are independent and identically distributed.
Under these conditions it suffices to maximize the product P (φn1) · · ·P (φnN) over
all possible values of γ. In other words, we can restate Equation 4.6 as maximizing
the product
γML = argmaxγ
[P (φn1) · · ·P (φnN)] , ∀γ
= argmaxγ
[N∑k=1
log [P (φnk)]
], ∀γ (4.7)
CHAPTER 4. PERSISTENT SCATTERER SELECTION 44
This estimator reduces to evaluation of the sum of probability terms that can
be precomputed for different values of γ and φ using the expressions given in
Equation 3.10 or Equation 3.12. We do not see a significant difference in the results
obtained using the two different signal models described in Chapter 3 and hence,
assume the Gaussian signal model (Section 3.3.2) in all further discussions due to its
closed form PDF expression (Equation 3.12).
We compare the maximum likelihood estimate γML from Equation 4.7 for each
of the candidate pixels against a pre-determined threshold value (γthr) - those
that exceed the threshold form the set of candidate PS pixels. The successful
implementation of our PS selection algorithm depends on the properties of our
estimator (Equation 4.7) and the choice of the threshold (γthr). We elaborate on
each of these aspects in the next few subsections.
4.4.1 Estimator properties
In this section, we describe some of the salient features of the MLPS estimator
(Equation 4.7) and show that it outperforms the conventional temporal coherence
estimator described in Equation 4.3. Replacing the temporal coherence estimator
with the MLPS estimator in the StaMPS framework leads to a significantly larger
number of PS pixels being correctly identified.
1. Random phase sequences
Figure 4.3 compares the ability of the temporal coherence estimator (Equa-
tion 3.15) and the MLPS estimator (Equation 4.7) to identify pixels with
random interferometric phase observations. The MLPS estimator has a lower
bias compared to the conventional coherence estimator. The estimates were
computed for a million randomly simulated phase sequences of length 40. The
tail of the histogram corresponding to the MLPS results (dotted lines) also drops
to zero faster than the conventional coherence results (solid line). Consequently,
the MLPS estimator identifies a larger percentage of random pixels as non-PS
pixels than the temporal coherence estimator.
CHAPTER 4. PERSISTENT SCATTERER SELECTION 45
Figure 4.3: Comparison of the ability of the conventional coherence estimator and theMLPS estimator to identify pixels with random interferometric phase. The MLPSestimator has a lower bias and drops to zero faster than the temporal coherenceestimator. The MLPS estimator successfully identifies a larger percentage of randomphased pixels as non-PS pixels than the temporal coherence estimator.
2. Effect of sample size
The MLPS estimator exhibits expected behavior when the number of input
samples for the estimator are changed (Figure 4.4). The bias in the estimates
increases as the number of available phase observations decreases. The variance
in the estimates also increases for fewer number of observations. From
Figure 4.4, it is clear that the MLPS estimator outperforms the conventional
coherence estimator even for smaller sample sizes.
3. Accuracy of the estimator
We define the accuracy of the MLPS estimator as it’s ability to identify
the true SCR used to simulate a sequence of phase observation according to
the gaussian signal model. Figure 4.5 shows the estimated SCR (γML) as a
function of true SCR. We simulated 100000 InSAR time-series corresponding to
25 interferograms for each SCR value (spacing of 0.05) for the Gaussian signal
CHAPTER 4. PERSISTENT SCATTERER SELECTION 46
Figure 4.4: The effect of sample size on the ability of the MLPS estimator toidentify random noisy pixels. The histograms of estimated correlations for simulatedrandom phase sequences of various lengths - (Top) MLPS estimator and (Bottom)Conventional coherence estimator. The experiment used to compute Figure 4.4 wasrepreated for 20 (dots), 30 (dots and dashes) and 40 (solid lines) interferograms.The bias in estimated coherence decreases with increased sample size for both theestimators.
model and used the estimator given in Equation 4.7 to generate the image. The
MLPS estimator (Equation 4.7) slightly underestimates the true SCR value but
the slope of the line is very close to one to ignore this disparity. We notice that
the estimated SCR never reaches zero due to the bias from the finite sample
size.
CHAPTER 4. PERSISTENT SCATTERER SELECTION 47
Figure 4.5: Estimates Most Likely SCR (γML) using Equation 4.7 as a function oftrue SCR value for 100000 simulate pixels, assuming a time-series consisting of 25interferograms and the Gaussian signal model. The theoretical unitary slope linecorresponding to an ideal estimator is also shown.
4.4.2 Threshold Selection
The set of candidate PS pixels depends on our selection of the proper γthr. If the
threshold is too high, many potential PS pixels are not included and the resulting
PS network is sparse. If the threshold is set too low, too many non-PS pixels are
included and the network contains many points that are not truly persistent. Our
threshold selection is currently rather arbitrary, because we cannot easily calculate
the scattering statistics for persistent pixels amid all possible background terrains.
If we could set a limit on the natural variation in the phase of each pixel, we could
readily select the threshold parameter γthr from the dependence of phase standard
deviation on γ (Figure 4.6). If we believe, for example, that we should see a natural
variation of a fifth of a cycle across all interferograms, then the threshold value is
about 2.
Alternatively, we could use one of the common radar detection criteria like
constant error rate (e.g. see Skolnik (2001)) to set our threshold. The error rate
CHAPTER 4. PERSISTENT SCATTERER SELECTION 48
Figure 4.6: Standard deviation of interferometric phase for the constant signal modeland the gaussian signal model. We can estimate the threshold (γthr) by setting alimit on the natural phase variation.
(ER) for a given threshold (γ0) can be numerically estimated using the equation
ER (γ0) =
∫ γ0
0
f (β) · P (γest > γ0|γ = β) · dβ +∫ inf
γ0
f (β) · P (γest < γ0|γ = β) · dβ (4.8)
The first term in both the integrals (f (β)) is the PDF of the SCR of the scatterers
in the scene that can be estimated from the data itself using the Equation 4.7. We
estimate the second term in both the integrals in the right hand side of Equation 4.8
by simulating sequences of interferometric phase residuals from the theoretical PDFs
(Equation 3.10 or Equation 3.12) using the MLPS estimator in Equation 4.7. For an
error rate of 5%, we can then estimate our threshold using
γthr = argminγ
ER (γ) , ∀ γ s.t. ER (γ) < 0.05 (4.9)
ER is often a misleading metric in practical applications, because the penalty of non-
selection of true PS pixels is less compared to the penalty of selection of non-PS pixels.
More importantly, selecing fewer non-PS pixels with random phase characteristics
CHAPTER 4. PERSISTENT SCATTERER SELECTION 49
significantly improves our ability to reliably unwrap the phase data and decrease
noise in our deformation estimates. Hence, a more appropriate metric would be the
random pixel acceptance rate (RPAR) defined by the rate of acceptance of pixels
with truly random or uniform phase distribution. Figure 4.7 shows the estimated
RPAR for both the signal models as a function of the SCR threshold (γthr) obtained
by simulating 100,000 sequences of random phase values in the interval [−π, π) and
estimating the corresponding γML using Equation 4.7. For a RPAR of 1%, we could
estimate the SCR threshold using the following equations.
RPAR (γ0) = P (γest > γ0|γ = 0) (4.10)
γthr = argminγ
[RPAR (γ)] ∀ γ s.tRPAR (γ) < 0.01 (4.11)
From Figure 4.7, we observe that for a SCR threshold of 1.8, the RPAR for both the
Figure 4.7: RPAR values for various SCR thresholds(γthr) for the constant andgaussian signal models, and amplitude dispersion method. At SCR value of 2, therandom pixel selection rate is less than 1% for the constant and gaussian signalmodels. RPAR for the amplitude dispersion method does not reduce to 1% till anSCR threshold of about 3.2.
signal models is less than 1%.
An SCR threshold value of 2 fulfilled all the three PS selection criterion described
in this section, for the data sets that we examined. Similar threshold values for SCR
CHAPTER 4. PERSISTENT SCATTERER SELECTION 50
have been used in other PS algorithms as well (Adam et al., 2003; Kampes and Adam,
2005).
4.5 An example application
We applied our PS pixel selection method to ERS data acquired over the San Francisco
bay area in two different regions: along the San Andreas fault near the SFO airport
region and along the Hayward fault near the Oakland-Alameda area. These areas
consist of a combination of urban and natural terrain clearly traversed by active
faults. Conventional InSAR studies combined with GPS observations (Burgmann
et al., 2000) measure the fault creep deformation signal well in the urban regions
along the Hayward Fault, with inconclusive measurements along the San Andreas
fault. Previous PS studies (Burgmann et al., 2006) conducted in this area also pick
out coherent scatterers in the urban areas well, but fail in identifying a reliable network
of scatterers in the natural terrain on the more rural sides of the faults.
We processed 19 descending scenes acquired by ERS-1 and ERS-2 between 1992
and 2000. We selected a scene from September 1995 as a master scene, based on
minimization of the perpendicular baseline and the temporal baseline, and generated
18 interferograms. The maximum perpendicular baseline was 260 m. Using the
maximum likelihood selection method, we can identify coherent pixels on the non-
urbanized sides of each fault, including where the Ferretti and StaMPS methods fail
to locate any PS pixels (See Figure 4.8). Figure 4.9 shows the observed average
deformation rates in mm per year and the locations of PS pixels for the San Andreas
Fault and Hayward Fault region. Creep of 3 mm/yr (LOS) shows clearly along the
Hayward fault, but no evidence of creep greater than possibly 1-2 mm/yr appears
along this segment of the San Andreas fault. Notably, in both these regions our
methods identified PS in regions where other methods have failed to identify any
coherent scatterers. Once we identify the proper set of PS pixels, we obtain a
deformation time-series by unwrapping the measured phases in three dimensions, two
CHAPTER 4. PERSISTENT SCATTERER SELECTION 51
spatial dimensions and one in time. We used the step wise-3D unwrapping method
described by Hooper and Zebker (2007).
−122.45 −122.4 −122.35 −122.3
37.5
37.55
37.6
37.65
−122.45 −122.4 −122.35 −122.3
37.5
37.55
37.6
37.65
−122.45 −122.4 −122.35 −122.3
37.5
37.55
37.6
37.65
−122.2 −122.16 −122.12 −122.08 −122.04
37.72
37.76
37.8
37.84
37.88
−122.2 −122.16 −122 .12 −122.08 −122.04
37.72
37.76
37.8
37.84
37.88
−122 .2 −122 .16 −122 .12 −122.08 −122.04
37.72
37.76
37.8
37.84
37.88
San Andreas Fault Area
Hayward Fault Area
Ferretti et al. (2000) Hooper et al. (2004) Maximum Likelihood
Figure 4.8: Binary mask of the identified PS for the SFO airport region (top) andOakland-Alameda region (bottom). Results from (left) Ferretti, (middle) Hooperand (right) maximum likelihood methods. The MLPS approach identifies a denserPS network in natural terrain than either of the published methods.
Examination of the PS pixels outside of the urban areas in both regions of
study shows that there are areas with spatially correlated phases distinct from the
average background velocities. While these could be areas of coherent motion such
as landslides, subsidence, or other localized movement, they can just as easily be the
product of phase unwrapping errors. At present we are unable to validate the absolute
accuracy of the phase unwrapping and this appears to be the limiting problem in the
application of PS methods to natural terrains. In Chapter 7, we provide a detailed
analysis of the time-series of many of the features observed in Figure 4.9.
4.6 Conclusions
The new maximum likelihood methods for the identification of candidate PS pixels
finds many more PS points than published algorithms. When applied to ERS-1
CHAPTER 4. PERSISTENT SCATTERER SELECTION 52
Figure 4.9: Comparison of PS identification algorithms for the SFO airport region(top) and Oakland-Alameda region (bottom). Results from (left) Ferretti, (middle)Hooper and (right) maximum likelihood methods. The MLPS approach identifies adenser PS network in natural terrain than either of the published methods.
Table 4.1: Comparison between different PS selection methods for two test regionsin the San Francisco Bay Area
SFO airport region Oakland-Alameda regionPermanent Scatterers R© Permanent Scatterers R©
(6800) (13237)StaMPS MLPS StaMPS MLPS
PS Candidates 325554 325554 485983 485983Pre-weeding 31395 50883 (62%)a 80469 108939 (35%)a
Selected PS 20012 32683 (63%)a 46181 62733 (35%)a
Common with StaMPS All 19568 (98%) All 45034 (98%)a indicates an increase.
and ERS-2 data acquired over the San Francisco bay area, our PS selection method
identified 62% (San Andreas Fault region) and 35% (Hayward Fault region) more
persistent pixels than Hooper’s (2004) algorithm, which in turn finds more PS pixels
than Ferretti’s original (2001) method. We find 97% of the pixels identified by Hooper
plus many more, notably in the vegetated areas in which both previous methods fail
to find sufficient persistent scatterers to form a usable network to infer deformation.
CHAPTER 4. PERSISTENT SCATTERER SELECTION 53
Thus the major advantage of the new method is that it allows us to find persistent
scattering pixels in vegetated and non-urban terrain, as shown here in the hills
west of the San Andreas fault near the SFO airport and the hills to the east of
Hayward fault. Very few coherent scatterers are identified here using the methods of
Ferretti et al. (2001) and Hooper et al. (2004), greatly limiting our ability to measure
deformation accurately. The PS pixels identified by the maximum likelihood method
are mostly a superset of those identified by published methods. Significant work
remains in the development of useful 3-dimensional phase unwrapping algorithms.
Phase unwrapping error is still the major source of uncertainty for PS-InSAR in
vegetated areas. The increased density of PS points provided by the maximum
likelihood selection method promises to make the unwrapping problem more tractable.
We have addressed the phase unwrapping problem in Chapter 5 with the development
of two new unwrapping algorithms suitable for PS-InSAR applications.
Chapter 5
Phase Unwrapping
In chapter 4, we described a new technique for identifying a dense network of
persistently scattering pixels. The key processing step in extracting the surface
deformation is the estimation of the unwrapped phase for this network of PS pixels.
In this chapter, we discuss the spatially sparse data problem, the three dimensional
phase unwrapping problem, and a new integer programming formulation for phase
unwrapping of multi-dimensional data.
Phase unwrapping is a key problem in many coherent imaging systems, including
time series synthetic aperture radar interferometry, which has two spatial and one
temporal data dimensions. The problem poses a harder challenge when the data
is spatially sparse, due to the availabilty of few reliable PS pixels over the area of
interest. We first present a simple method to reduce sparse data sets to equivalent
regularly sampled data sets for two-dimensional data. We then describe the edgelist
phase unwrapping algorithm which builds on the concepts of the minimum cost flow
(MCF) formulation of Costantini (1998). The MCF phase unwrapping algorithm
describes a global cost minimization problem involving flow between phase residues
computed over closed loops. In our approach, we replace closed loops by reliable
edges as the basic construct, thus leading to the name “edgelist”. Our algorithm has
several advantages over current methods :
• It simplifies the representation of multi-dimensional phase unwrapping.
54
CHAPTER 5. PHASE UNWRAPPING 55
• It can incorporate data from external sources, like GPS, where available to
better constrain the unwrapped solution.
• It treats regularly sampled or sparsely sampled data alike.
It is thus particularly applicable to time series InSAR where data are often irregularly
spaced in time, and individual interferograms are corrupted with large decorrelated
regions. We show that, similar to the MCF network problem, our formulation also
exhibits total unimodularity (TUM) which enables us to solve the integer program
using efficient linear programming tools. We then apply our method to a PS-InSAR
dataset from the creeping section of the Central San Andreas Fault.
5.1 Introduction
Phase unwrapping as used in InSAR geodesy is the reconstruction of absolute phase
from measured phase known only modulo 2π on a finite grid of points. Many methods
have been developed for unwrapping SAR interferograms (Hunt, 1979; Goldstein
et al., 1988; Ching et al., 1992; Ghiglia and Romero, 1996; Pritt, 1996; Flynn, 1997;
Zebker and Lu, 1998; Costantini, 1998; Chen and Zebker, 2000), however phase
unwrapping is a key step in many other coherent imaging techniques as well. Most
of these methods primarily focus on regularly sampled two dimensional data sets.
Interferograms, especially in time series analysis, are often irregularly sampled in both
space and time, so that existing algorithms do not always properly unwrap the data.
In this chapter, we develop a method to address the most general form of the phase
unwrapping problem - unwrapping sparsely distributed, multi-dimensional wrapped
phase data. Since we focus on applying our new technique to InSAR phase data, we
restrict our discussion to conventional two dimensional (single interferogram) InSAR
data and multi-temporal InSAR (Persistent Scatterer and Short Baseline methods)
three dimensional data sets.
The performance of any phase unwrapping algorithm depends on our ability to
estimate the phase gradient between two adjacent samples in our data set. The
CHAPTER 5. PHASE UNWRAPPING 56
basic assumption in all phase unwrapping problems is that the underlying continuous
unwrapped phase function is well sampled in every dimension to enable us to
reconstruct it from wrapped phase measurements except at a finite, relatively small,
number of discontinuities.
The existence of discontinuities in a data set produces path dependent inconsis-
tencies or residues (Goldstein et al., 1988). The branch cut algorithm (Goldstein
et al., 1988) and its derivatives are very popular approaches to phase unwrapping,
due to their ease of implementation. Hooper and Zebker (2007) successfully extended
the idea of the shortest branch cut to the 3D phase unwrapping problem. Least
squares (Hunt, 1979) and FFT-based (Ghiglia and Romero, 1994) phase unwrapping
algorithms are computationally efficient but tend to distribute unwrapping errors
globally instead of restricting errors to a small set of points. Ghiglia and Romero
(1996) suggested a general Lp norm objective function for phase unwrapping problems
and argued that L0 and L1 norm solutions would produce fewer errors than the
traditional least squares solution.
Costantini (1998) developed the first network programming formulation to solve
the regularly sampled two dimensional (2D) phase unwrapping problem using an L1
norm minimum cost flow (MCF) approach. Costantini and Rosen (1999) adapted
the algorithm to solve the irregularly sampled 2D phase unwrapping problems using
Delaunay triangulation (Barber and Huhdanpaa, 2009). The performance of the
network-programming-based unwrapping algorithms for 2D data sets was further
improved by the development of an iterative L0 norm approximation algorithm (Chen
and Zebker, 2000) and its application in combination with statistical cost functions
(Chen and Zebker, 2001).
We propose a new minimum L1 norm formulation that is more flexible than
the original MCF formulation, allowing us to impose additional constraints on the
unwrapped solutions based on available a priori information. It also can be easily
extended to analyze multi-dimensional data sets. Here, we briefly review conventional
residue-based phase unwrapping algorithms (Section 5.2). We then describe our
CHAPTER 5. PHASE UNWRAPPING 57
new algorithm in Section 5.5. In Section 5.6, we discuss the ability to incorporate
additional geodetic observations as constraints in our new algorithm. We apply
the method to a time series of data acquired over the creeping section of the San
Andreas Fault in Section 5.7. Finally, we discuss implementation and suggest possible
improvements.
5.2 Residue-based unwrapping techniques
The primary force motivating the development of residue-based phase unwrapping
algorithms is the notion that all the phase discontinuities in a two-dimensional data set
are due to linear discontinuities. Goldstein et al. (1988) observed that the endpoints
of these linear discontinuities can be identified using a discretized curl measure, also
known as a residue, as the total phase gradient around the end points does not add up
to zero. Example of such closed loops with zero, +1 and −1 are shown in Figure 5.1.
The residues are assumed to be located at the centroids of the corresponding loops.
In the original residue-cut algorithm (Goldstein et al., 1988), a non-zero residue
is located and connected to the nearest non-zero residue, irrespective of the sign.
The process of locating and adding non-zero residues to this tree continues until the
tree is neutralized. Once neutralized the whole process is repeated again starting
from another non-zero residue. When all the residues have been neutralized, the
wrapped phase is integrated along any path like a one-dimensional problem (flood-fill
algorithm) without integrating over the branch cuts or linear discontinuities. The
main aim of the algorithm was to reduce the total length of linear discontinuities
imposed to unwrap the data set.
Chen and Zebker (2000) showed that the shortest branch-cut problem reduces
to the minimum Steiner tree problem and is NP-hard. They observed that the
minimum spanning tree (MST), which is the shortest path connecting all the residues,
is a good approximation of the minimum steiner tree and overcomes the problem
of disconnected regions produced by the original branch-cut algorithm. These
CHAPTER 5. PHASE UNWRAPPING 58
Figure 5.1: Determination of phase residues in a regularly sampled two-dimensionalinterferogram. [·] represents the nearest integer function. Loops with non-zeroresidues represent the end-points of linear discontinuities marking phase cycle jumps.
branch-cut algorithms perform well in regions of high coherence, but fail to perform
satisfactorily in noisy decorrelated sections of inteferograms (Chen and Zebker,
2000). These algorithms inherently cannot accomodate quality measures and external
information in placing these branch cuts to produce more reliable solutions.
5.2.1 Network programming approach
Costantini (1998) exploited the equivalence between the constrained minimum
spanning tree problem and the minimum cost flow (MCF) problem to reformulate
the problem as a network programming problem. The MCF formulation allows for
quality measures in form of cost functions to automatically direct the placement of
branch cuts.
The MCF formulation explicitly allows us to solve for the integer phase cycles
(Kij) that need to be added to the each edge (i, j) of the unwrapping grid to produce
a consistently unwrapped solution. If we introduce non-zero cost functions (Cij)
CHAPTER 5. PHASE UNWRAPPING 59
associated with adding a unit phase cycle over each edge (i, j), the two-dimensional
phase unwrapping problem can be formulated as a minimum cost flow problem as
follows (Costantini, 1998):
Minimize∑
∀(i,j)∈ECij · ||Kij||
Subject to
KAB +KBC +KCD +KDA =
[ψA − ψB
2π
]+
[ψB − ψC
2π
]+[
ψC − ψD
2π
]+
[ψD − ψA
2π
]∀ simple closed rectangles (A,B,C,D)
Kij is an integer ∀(i, j) ∈ E (5.1)
where the right hand side of the constraint equation corresponds to the residue
enclosed by each loop (Figure 5.1), || · || refers to the L1 norm and E represents
the set of all edges belonging to a simple rectangular grid. The additional integer
cycles Kij that need to be added is also referred to as the integer flow on the edge
(i, j). The cost functions Cij are used to direct the placement of phase cycle jumps
using quality measures derived from the data itself.
5.3 Reduction of sparse data on a regular grid
New multiple pass InSAR techniques often produce a sparse set of points that
represent the only reliable measurements of line of sight (LOS) displacement
(Chapter 4). PS points are often widely spaced and irregularly sample the deformation
pattern. Many current phase unwrapping methods such as SNAPHU (Chen and
Zebker, 2001) are not designed to use arbitrarily spaced data, and thus their major
benefits of constraining possible and optimal solutions cannot be realized. In this
section, we describe a resampling method that enables us to use popular and freely
available regular grid 2D phase unwrapping software to unwrap sparse data sets
CHAPTER 5. PHASE UNWRAPPING 60
effectively. Numerous approaches to phase unwrapping on a regular grid have been
suggested in the recent past. These unwrapping approaches rely on the concept of
residues (Goldstein et al., 1988) about closed loops, using estimates of integrated
phase gradients to identify areas of inconsistency. Network programming approaches
(Chen and Zebker, 2000; Costantini, 1998; Chen and Zebker, 2001) have so far proven
to be the most effective of these. These methods have also been extended to unwrap
sparse 2D data sets (Costantini and Rosen, 1999).
For reducing the sparse data to a regular grid, we adopt the concept of residues
defined by closed loops of triangles from a non-overlapping triangulation of the given
set of sparse data points. The data are then resampled to a regular Cartesian grid
and unwrapped with existing solvers. All the results shown here were produced
using the Statistical-cost Network flow Algorithm for Phase Unwrapping (SNAPHU)
software (Chen and Zebker, 2001). However, any unwrapping algorithm may be used
once the sparse 2D unwrapping problem is transformed to a regularly-spaced grid
unwrapping problem with our method. This approach for reducing sparse unwrapping
problem to a regular grid unwrapping problem was also developed independently by
Hooper (Hooper, 2009), who implemented it in the StaMPS software package for PS
processing.
5.3.1 Definitions
Let S represent a set of distinct points on a 2D plane where the wrapped phase values
are known. Often, these are a subset of points on a regular rectangular 2D grid as
represented in Figure 5.2 by distinct symbols enclosed in circles. We now define two
real functions ψ(s) and φ(s) for each point , such that
ψ (s) = φ (s) + 2πn (s) (5.2)
where n(s) is an integer such that ψ(s) ∈ [−π, π) . The functions ψ(s) and φ(s) are
the wrapped and unwrapped phase functions respectively. Any useful unwrapping
CHAPTER 5. PHASE UNWRAPPING 61
Figure 5.2: Reduction of a sample sparse data set to a regular 2D grid. Symbolsenclosed in circles represent the original sparse data points. Solid lines anddashed lines represent the corresponding Delaunay triangulation and Voronoi diagramrespectively. Other data points in the regular grid have been filled in with symbolsusing the nearest neighbor interpolation. Note that the Voronoi diagram correspondsto the boundaries of the interpolated regions.
approach estimates n(s) from a given ψ(s) such that the resultant unwrapped phase
function φ(s) is irrotational, i.e curl of φ(s) reduces to zero. We also use two
geometrical concepts called the Voronoi diagram and the Delaunay triangulation
of the set of points S (Barber and Huhdanpaa, 2009). The Voronoi diagram(S)
and the Delaunay triangulation(S)
are represented respectively by dashed and
solid lines in Figure 5.2. The Voronoi diagram(S)
is a continuous version of the
regularly interpolated the grid. The Voronoi diagram and the Delaunay triangulation
are closely related and it can be shown that in graph theory terminology,(S)
is
the dual graph of(S)(Barber and Huhdanpaa, 2009). This property will be used
below to relate the regular grid of nearest neighbor interpolated data to the Delaunay
triangulation of the sparse data points.
The residues computed on S of the set of vertices S are analogous to residues
computed on a regular grid, in this case each triangle represents a non-overlapping
closed loop. In existing sparse 2D unwrapping approaches (Costantini and Rosen,
1999; Hooper and Zebker, 2007), residues are computed on each of the triangles and
CHAPTER 5. PHASE UNWRAPPING 62
assumed to be located at the centroid (point of intersection of perpendicular bisectors
of the edges) of triangles. In these methods, a branch cut algorithm or a maximum
flow approach is then used to neutralize the residues, and the phases are derived from
a filling algorithm that grows the phase field while avoiding inconsistent paths (see
Goldstein et al. (1988) for a discussion).
5.3.2 Geometrical reduction to regular problem
We now show that we can reduce the sparse unwrapping problem to a regular 2D
unwrapping problem using a nearest neighbor algorithm. We first interpolate the
sparse phase data onto a regular grid using the smallest Euclidean distance nearest
neighbor rule as described below. Co-circular vertices, or points that lie on the
same circle, amongst the sparse data points can produce a non-unique Delaunay
triangulation. We overcome this problem by perturbing or joggling the inputs S,
by addition of small random numbers (Barber and Huhdanpaa, 2009). This ensures
a unique triangulation(S)
and a convex Voronoi diagram(S). Efficient search
algorithms (Mount and Arya, 2010) are then used to determine the nearest neighbor
to a given arbitrary point from among a set of measured data points. An example is
shown in Figure 5.2. We refer to all interpolated data points that were not part of the
original sparse data set as replicas. These are represented by the symbols that are not
enclosed in circles in Figure 5.2. Close inspection makes it clear that the boundaries
of the nearest neighbor interpolated data approximates the Voronoi diagram, which
is also shown by dotted lines.
Next, we show that the phase residues are conserved in the nearest neighbor
interpolated regular 2D image. Figure 5.3 illustrates all the possible types of loops
that we encounter in the interpolated regular grid. For simplicity, we only show the
region within one of the Delaunay triangles and use distinct symbols to represent the
three vertices of the triangle. From Figure 5.3, the following can be inferred:
1. Loops formed in regions consisting of the same phase values (Figure 5.3a) do
not produce any residues.
CHAPTER 5. PHASE UNWRAPPING 63
(a) (b)
(c) (d)
(e)
Figure 5.3: This figure shows all possible types of loops that can be observed in aregular 2D image obtained by nearest neighbor interpolation. Distinct symbols areused to identify replicas of different sparse data points. Only the loops in (d) and (e)can produce a non-zero residue.
CHAPTER 5. PHASE UNWRAPPING 64
2. Loops formed on the boundaries with two points of each kind (Figure 5.3b)
do not produce residues, as we traverse the boundary twice but in opposite
directions when going around the loop. The residue equation in Figure 5.1
evaluates to zero.
3. Loops formed on the boundaries with 3 points of one kind and one of another
(Figure 5.3c) do not produce any residues. Again, we traverse the boundary in
opposite directions and residue equation in Figure 5.1 evaluates to zero.
4. The loop containing the circumcenter, where at least one value corresponding
to each vertex of the Delaunay triangle is present (Figure 5.3d) can produce
a residue. The value of this residue is same as that of the original Delaunay
triangle. Also, there can exist only one such loop within the triangular region
as the circumcenter of each triangle is a unique point.
5. The residues of both the Delaunay triangles (Figure 5.3e) are co-located at
the center of the square and the sum of the residues is same as that of the
original square. Loops of this kind prevent an exact reduction of this sparse
unwrapping problem to a regular unwrapping problem. Details are provided in
the next section.
6. Hence, all the residues in the sparse image are conserved and are localized at
the circumcenters of the Delaunay triangles.
We have thus shown that we can geometrically reduce the sparse unwrapping
problem to a regular 2D problem, and that there is a unique mapping resulting from
the nearest neighbor algorithm. Since the data is now defined on a regular grid, we
can solve the problem using conventional unwrapping techniques. This is strictly true
only if we ignore the cost functions for the edges formed during our original Delaunay
triangulation. In the next section, we describe the equivalence of branch cuts in the
original and the transformed problems. We also describe the changes required in the
cost functions for an almost equivalent reduction of the sparse unwrapping problem
to a regular 2D problem.
CHAPTER 5. PHASE UNWRAPPING 65
5.3.3 Branch cuts, flows and cost functions
If replicas of s1 and s2 are connected by at least an edge on the interpolated regular
grid, they are connected in the Delaunay triangulation (S) . This is a direct
consequence of the duality relationship between S and S. Since we approximate
the Voronoi diagram using a discrete regular grid, the inverse relationship is not
true. Figure 5.3e shows an instance where the points on the diagonal (diamond and
triangle) of the rectangle are connected in S but not on the regular grid. Almost
co-circular points often produce Voronoi boundary line segments that are completely
enclosed within a unit square, leading to loss of the edges connecting the diagonals. In
terms of geometry, this implies that the regular grid unwrapping problem corresponds
directly to the sparse unwrapping problem defined using residues computed over a
combination of triangles and quadrilaterals (except the lost edges, similar to merged
facets in (Barber and Huhdanpaa, 2009)) rather than the original sparse unwrapping
problem defined over S.
Also, since we preserve the geometry in terms of location of the residues at the
circumcenters of the triangles, the branch cuts occupy the same physical location in
the sparse and filled data sets. In other words, if the edge between s1 and s2 is marked
by a branch cut in(S), then all the edges between replicas of s1 and s2 will also be
marked by the branch cut. An example is provided in Figure 5.4. If the edge between
a diamond and a triangle (enclosed in circles) is crossed by a branch cut (dashed line)
in the original sparse problem, all the edges between the diamonds and triangles are
also crossed by the branch cut in the regular problem.
We arrive at the same conclusion using the arc network flow algorithms following
similar reasoning. In the previous section, we proved that three of the five different
types of loops cannot produce a residue. From property (1) in section 5.3.2, there
cannot exist a non-zero flow or a cycle jump on an edge connecting the replicas of the
same sparse point as this would imply that replicas of the same points are separated
by a nonzero multiple of 2π, thus producing an inconsistency. Hence, any flow is
restricted to edges connecting points with distinct phase values. From properties (2)
CHAPTER 5. PHASE UNWRAPPING 66
Figure 5.4: This example shows that a branch cut (dashed line) connecting thecircumcenters of two adjacent Delaunay triangles intersecting the edges betweenvertices represented by a diamond and a triangle, also disconnects all edges on theregular grid that connect replicas of the diamond and the triangle.
and (3), we can infer that if there exists a flow on the arc connecting replica points
of s1 and s2, all arcs connecting the replicas of s1 to those of s2 will carry the same
flow. Thus, in either unwrapping approach, the cost function needs to account for
the number of such edges between replicas of s1 and s2. We therefore define the cost
functions to be used on the nearest neighbor interpolated data on the regular grid as
Creg (s1, s2) =Csparse (s1, s2)
n (s1, s2)(5.3)
where Creg (s1, s2) is is the cost function for the edge joining a replica of s1 and a
replica of s2, Csparse (s1, s2) is the cost function for the edge joining points s1 and s2
on the Delaunay triangulation and n (s1, s2) is the number of edges between replicas
of s1 and s2 on the regular grid. The edges connecting the replicas of the same sparse
data point cannot sustain any flow because of reasons mentioned in section 5.3.2. In
our work, the cost function of such edges is set to a large value.
The objective function of the regular unwrapping problem does not contain terms
corresponding to the edges lost in the interpolation process. By comparing objective
functions, we can again conclude that the regular unwrapping problem is equivalent to
CHAPTER 5. PHASE UNWRAPPING 67
a sparse unwrapping problem defined using residues computed over a combination of
triangles and quadrilaterals, as described above. The number of lost edges is directly
related to the distribution of points in S and typically constitutes a very small fraction
(about one in a thousand) of all the edges in S. Often, if the data are well sampled,
they do not carry any flow across them and do not contribute to the overall objective
function. Hence, the regular unwrapping problem is almost equivalent to the sparse
unwrapping problem defined over S.
5.3.4 Examples
We applied our method to two data sets, one simulated and one real. We
first generated a synthetic interferogram of a vertical right-lateral strike slip fault
and descending right-looking geometry over a regular grid using Okada’s surface
deformation model (Okada, 1985), and then randomly selected 10% of the points
from the regular grid to form a sparse data set. Figure 5.5 shows the simulated
sparse interferogram, including the fault trace in black and the result of unwrapping
using L1-norm weights in SNAPHU. No additional noise was added to the data and
hence we used uniform weights on the original Delaunay triangles for unwrapping.
For a test using actual data, we produced a second interferogram from data of the
2003 Bam earthquake in Iran as measured using the SAR instrument on the Envisat
satellite (Funning et al., 2005). We randomly selected 10% of the points in this
case to simulate a sparse interferogram. Figure 5.6 shows the interferogram and its
corresponding unwrapped solution obtained with SNAPHU, in a manner that makes
it easy to compare the images directly with Figure 4 (b,d) from Funning et al. (2005).
The solution of our synthetic data set precisely matches the original data in the
simulated interferogram. The sparse-data Bam solution is geophysically plausible
but underestimates the total deformation reported by Funning et al. (2005). This is
a direct consequence of the sparseness of reliable measurements and the resulting loss
of residues or cycle jumps in regions of large deformation gradients. The accuracy
of our method and the ability to handle varying degrees of data sparseness depends
CHAPTER 5. PHASE UNWRAPPING 68
Figure 5.5: (a) Simulated interferogram of a right lateral strike slip fault with 10% ofthe randomly selected points from the complete interferogram. (b) The unwrappedinterferogram using SNAPHU and L1 norm weights. All the white regions in theimage indicate areas where no data is available.
on how the solver retrieves the missing phase cycles using the residues computed on
the Delaunay triangulation. Hence if too few data points are available, the solution
will be poor. The resampled-data problem can often be solved very quickly because
most of the potential residues on the regular grid are likely to be zero due to reasons
described in section 5.3.2. We have shown here that the solver in SNAPHU works
well for the two cases we have used, but any solver could be combined with a nearest
neighbor interpolator to unwrap sparse data.
Our method can also be used with existing regular grid 3D unwrapping software, if
available, to unwrap such data sets. In addition, our method does not need expensive
commercial solvers or complicated pre-processors and can be easily implemented with
existing regular grid solvers. Note that we do not claim that our method necessarily
outperforms other unwrapping methods, it simply makes the sparse unwrapping
problem feasible using many existing codes. The result of our transformation, applied
to sparse data, is almost the same as the solution obtained using residues computed
CHAPTER 5. PHASE UNWRAPPING 69
Figure 5.6: (a) Envisat ASAR interferogram of the Bam earthquake. 10% of thepoints were randomly selected from the complete interferogram. (b) The unwrappedinterferogram using SNAPHU and deformation mode cost function. This can bedirectly compared with Figure 4(b,d) in Funning et al. (2005).
on Delaunay triangles and the basic algorithm implemented by the software. The
examples shown in this section use only 10% of the interferograms which is almost 10
times as dense as conventional PS networks. However, these interferograms also have
significantly larger deformation signatures than in regions that we have applied our
PS-InSAR techniques (Chapter 6 and Chapter 7) to. Our unwrapping method works
effectively on sparser data sets that are characterized by slower deformation fields.
5.4 Three-dimensional unwrapping
So far, we have discussed the two-dimensional phase unwrapping problem. Time-
series InSAR allows us to sample the deformation signal as a function of time, albeit
sampled irregularly. Hence, we need a different approach to incorporate this addi-
tional information in unwrapping these three-dimensional data sets. Huntley (2001)
extended the idea of branch-cuts to three dimensional data sets and postulated that
CHAPTER 5. PHASE UNWRAPPING 70
residues connect to form loops or surface discontinuities in the 3D case. Subsequently,
numerous branch-cut algorithms based on this idea have been developed for medical
applications (Huntley, 2001; Cusack and Papadakis, 2002). Hooper et al. (2007)
generalized the framework for time-series InSAR applications and argued that similar
to shortest length linear discontinuities in the 2D case, residues are connected by
surfaces with the smallest area in the 3D case. The MCF approach also has been
extended to three dimensions for application with time-series InSAR techniques
(Costantini et al., 2001; Pepe and Lanari, 2006).
However, as we will show later in this chapter, residue-based unwrapping
techniques are inflexible and cannot accomodate a priori information to constrain
the unwrapped solutions. We developed the edgelist phase unwrapping algorithm
to address some of the shortcomings of residue-based unwrapping algorithms and to
allow us to unwrap spatially-sparse data sets, where the estimates of residues are less
reliable due to the greater spatial separation of coherent phase estimates.
5.5 Edgelist phase unwrapping formulation
The edgelist phase unwrapping formulation addresses some of the limitations of the
residue-based MCF formulation when handling spatially sparse time-series InSAR
dataset. The edgelist formulation is inspired by the dual of the minimum cost flow
problem (Ahuja et al., 1999). In this section, we present the mathematical formulation
for the edgelist phase unwrapping algorithm.
Let V represent the set of points (|V | = N) on which the set of measured wrapped
phase values,{ψi} where i ∈ (1, · · · , N) , is defined. Let E define the set of edges
consituting the unwrapping grid (|E| = M) on V such that, for every edge(i, j) ∈ E,
i < j. Thus G := (V,E) represents a directed graph which will be used to estimate the
set of unwrapped phase values, {φi} where i ∈ (1, · · · , N). For simplicity, we assume
that the graph G represents the Delaunay triangulation of the set of points V and
CHAPTER 5. PHASE UNWRAPPING 71
E represents the set of the edges of the triangulation. The wrapped and unwrapped
phase values at each point in V are related by
φi = ψi + 2πni where i ∈ (1, · · · , N) (5.4)
where ni represent the integer number of cycles that must be added to each point
of the wrapped function to obtain the unwrapped function. These variables can be
interpreted as node potentials (Ahuja et al., 1999). For every edge (i, j) in E , we
define a new variable Kij such that
nj − ni +Kij =
[ψi − ψj
2π
](5.5)
Here [·] represents the nearest integer function and Kij represents the integer flow
along the directed edge (i, j) and is equivalent to variables K1 and K2 in (Costantini,
1998). As in the original MCF formulation, we associate a non-negative convex cost
function f (Kij) with the integral flow on every edge (i, j]) ∈ E . We can then state
our minimum cost flow problem as
Minimize∑
∀(i,j)∈Ef (Kij) (5.6)
Subject to
nj − ni +Kij =
[ψi − ψj
2π
](5.7)
ni ∈ Integer ∀i ∈ (1, · · · , N) (5.8)
Kij ∈ Integer ∀ (i, j) ∈ E (5.9)
Our formulation differs from that of Costantini (1998) in that the basic unit in our
algorithm is an edge on the unwrapping grid instead of a closed loop, hence the name
“edgelist”. This formulation also reduces to a variation of the convex cost integer dual
network flow problem (Ahuja et al., 1999). Lagrangian relaxation and cost scaling
algorithms can be applied to solve the general convex minimum cost flow problem
CHAPTER 5. PHASE UNWRAPPING 72
(Equations 5.6-5.9) (Ahuja et al., 1999). Following Costantini (1998), we restrict our
discussion to minimum L1 norm solutions for ease in implementation using LP solvers.
The salient features of the edgelist formulation include
1. The edgelist formulation reduces to the original MCF formulation if every edge
of the Delaunay tessellation is included as a constraint (See Fig 5.7). The
constraint of our phase unwrapping formulation when applied to the edges of a
loop produces a loop constraint in the original MCF formulation.
2. The edgelist formulation does not distinguish between 2D and 3D data,
where the third dimension is generally the time dimension in a series of
interferograms. Each phase measurement is treated as a distinct vertex of the
graph. Consequently, more variables and constraints are needed to completely
define a problem. Table 5.1 provides a comparison of the resources for when
each algorithm is applied to a 2D unwrapping problem.
3. The constraint matrix of the edgelist formulation is a total unimodular matrix
(TUM, see Section 5.5.1) and the right hand side of the constraints in
Equation 5.7 is an array of integers. Similar to the MCF and other TUM integer
programming problems, the edgelist formulation can also be exactly solved as
a linear program (LP) when the associated objective functions minimize the L1
norm (Hoffman and Kruskal, 1965).
4. The edgelist formulation can readily incorporate other geodetic measurements
such as GPS or leveling data as additional constraints without affecting the
TUM property of the constraint matrix. This is discussed in detail in section 3.
We alter the general edgelist formulation (Equations 5.6-5.9) to allow minimum
L1 norm solutions using LP solvers by transforming the L1 problem into a linear
program. Define two new sets of non-negative variables Pij and Qij , such that
Kij = Pij −Qij ∀ (i, j) ∈ E (5.10)
CHAPTER 5. PHASE UNWRAPPING 73
Figure 5.7: Comparison of the new edgelist formulation and the original MCFformulation. We obtain the constraints of the MCF formulation by summing upconstraints of the edgelist formulation.
Table 5.1: Parameters needed to represent a regular grid 2D unwrapping problem ofsize M x N pixels using both the edgelist and MCF formulations.
Minimum Cost Flow EdgelistDimensions of interferogram M ×N M ×NNumber of variables 4MN − 2M − 2N 5MN − 2M − 2NNumber of constraints MN −M −N + 1 2MN −M −NNon-zero constraint matrix entries 8MN − 8M − 8N + 8 8MN − 4M − 4N
The resulting constraint matrix is also TUM (See Section 5.5.1). Hence, the integer
program (Equations 5.6-5.9) can be solved exactly using its LP relaxation, where all
the variables are allowed to be real valued. The solution to our phase unwrapping
problem is thus obtained by solving the linear program
CHAPTER 5. PHASE UNWRAPPING 74
Minimize∑
∀(i,j)∈ECij · (Pij +Qij) (5.11)
Subject to
nj − ni + Pij −Qij =
[ψi − ψj
2π
](5.12)
ni ∈ Integer i ∈ (1, · · · , N) (5.13)
Pij, Qij ≥ 0 and real ∀ (i, j) ∈ E (5.14)
The node potentials and flows are solved simultaneously. As a result, the final
unwrapping step involves simple addition of the node potentials to the wrapped values
(Equation 5.4).
5.5.1 Total unimodularity
In this section, we prove the total unimodularity property of the constraint matrix in
the edgelist formulation. We use the following properties to prove that the edgelist
formulation can be solved using a real valued relaxation (Hoffman and Kruskal, 1965):
1. The incidence matrix (Gn) of a directed graph G is a totally unimodular matrix
(TUM).
2. If matrix A is a TUM and I is an identity matrix of appropriate dimensions,
then the following matrices are also TUM: −A, AT , [A, I], [A,−A].
3. Let A be a m × n TUM matrix. Then the following polyhedrons are integral
for any vector b of integers:
x ∈ Rn : A · x ≤ b
CHAPTER 5. PHASE UNWRAPPING 75
Revisiting the constraints for the edgelist formuation (Equation 5.7), the corre-
sponding constraint matrix can be written as
[Gn | I] ·⎡⎣ n
K
⎤⎦ = b (5.15)
where n represents the node potentials and K represents the flow variables. From
properties (1) and (2) above, A = [Gn | I] is TUM. Equation 5.15 can be reduced to
the form described in property (3) above as
⎡⎣ A
−A
⎤⎦ ·
⎡⎣ n
K
⎤⎦ ≤
⎡⎣ b
−b
⎤⎦ (5.16)
where all the entries of matrices A, b, n and K are integers.
Using property (3) above, we prove that our integer program (Equations 5.6 - 5.9)
can be solved exactly by its real-valued relaxation, as all the corresponding extreme
points are integers. A similar argument can be provided to show the LP relaxation
(Equations 5.11 - 5.14) also has a totally unimodular constraint matrix and solves
the integer problem of interest exactly.
5.5.2 Example
We illustrate the solution to our formulation using a LP solver on the example 3× 3
square grid of Figure 5.8. The index of the pixel and the corresponding wrapped
phase value are also shown in the format. A typical LP solver starts with a basic
feasible solution and minimizes the objective function by altering the solution in
the direction of highest negative gradient or the edge of the polytope with highest
negative gradient. The linear program edgelist formulation (Equations 5.11 5.14)
always has a basic feasible solution corresponding to all ni = 0, Pij =([
ψi−ψj
2π
]> 0
)and Qij =
([ψi−ψj
2π
]< 0
). In case of the example in Figure 5.8 and uniform costs
for all edges, the basic feasible initial solution corresponds to all variables set to zero,
CHAPTER 5. PHASE UNWRAPPING 76
Figure 5.8: An example 3 x 3 grid with pixel index (i) and the corresponding wrappedphase values (ψi) shown in (i, ψi) format. The optimal solution corresponds to variablen5 being set to one and all flow variables (Kijs) set to zero.
except P25, P45, P56 and P58 which equal one. The LP solver alters the solution
in the direction of the maximum negative gradient and the final optimal solution
corresponds to all variables being set to zero except n5 which equals one.
It is evident from Equations 5.11-5.14 that the solution is dependent on the
unwrapping grid (G) used to define the unwrapping problem. We used Delaunay
triangulations to obtain simple non-overlapping surface elements for the reliable
points in each interferogram (Costantini and Rosen, 1999). In case of three
dimensional PS-InSAR and SBAS data sets, these triangulations are replicated
in space for each individual interferogram and form rectangular facets in time,
following Hooper and Zebker (2007). The edgelist algorithm allows us to evaluate
the unwrapped solution on customized unwrapping grids by defining the set of edges
(E) appropriately. The optimal solution of Equations 5.11-5.14 also depends on the
values of cost functions (Cij) for the edges. These cost functions allow us to constrain
the flow across the edges and is a feature that is common to both the original MCF
and the edgelist formulations. For our time-series InSAR applications, we use the
wrapped phase data itself to estimate spatial and temporal phase gradients for use in
combination with the deformation mode statistical cost functions developed by Chen
CHAPTER 5. PHASE UNWRAPPING 77
and Zebker (2001). In section 5.7 we show an example where these cost functions are
modified to include a priori information about the area being analyzed.
5.6 Incorporating other geodetic measurements as
constraints
Often, complementary geodetic measurements such as GPS networks or leveling
surveys are available in addition to frequent SAR acquisitions over regions of interest.
This information can be used to direct edges in branch cut based algorithms or adjust
cost functions in a network programming method for unwrapping interferometric
phase, constraining the result to reflect these additional data. In branch cut
algorithms, incorporating such changes involves defining and implementing a multi-
criteria or Pareto optimal spanning tree problem (Zhou and Gen, 1997; Knowles
and Corne, 2001). In the original MCF formulation, constraints based on additional
observations can be defined but at the cost of violating the TUM property of the
constraint matrix (Section 5.6.1). Violation of the TUM property renders the problem
unsolvable exactly using LP solvers.
In the case of the edgelist formulation, if alternate geodetic measurements are
available at points p and q, we introduce a new edge between the points (see
Figure 5.9) and a new constraint in the formulation
np − nq = ΔNp,q −[ψp − ψq
2π
](5.17)
where ΔNp,q is the expected number of unwrapped phase cycles between p and q.
The new constraint adds additional entries to the original node but retains the TUM
structure (Section 5.5.1). If the points p and q were already connected in the original
unwrapping grid (G), the corresponding constraint for the edge in the unconstrained
formulation (Equation 5.12) can be replaced by the new equation (Equation 5.17).
Independent geodetic estimates of line of sight displacement for L vertices in V
CHAPTER 5. PHASE UNWRAPPING 78
Figure 5.9: An example 3 x 3 grid with pixel index and the corresponding wrappedphase values shown in (i, ψi) format. The optimal solution corresponds to variable n5
being set to one and all flow variables (Kij’s) set to zero.
will allow us to construct C (L, 2) = L·(L−1)2
additional equations to constrain our
unwrapped solutions.
Simpler constraints of the form
0 ≤ Pij ≤ u ∀ (i, j) ∈ E (5.18)
0 ≤ Qij ≤ u ∀ (i, j) ∈ E (5.19)
where u is a positive integer, can be further applied to reduce the solution space.
These constraints do not affect the TUM property. The primary advantage of the
edgelist formulation is that it provides us with controls over every data point (ni) and
every edge (Pij, Qij) , as opposed to control over the edges alone in case of the MCF
formulation. Both these properties can be suitably exploited to solve challenging
unwrapping problems as shown in Section 5.7.
CHAPTER 5. PHASE UNWRAPPING 79
5.6.1 Total unimodularity
In this subsection, we prove that the constraint matrix for the edgelist formulation
continues to be TUM after inclusion of external constraints of the form shown in
Equation 5.17, whereas the MCF formulation no longer remains TUM on inclusion
of external constraints. We use the following property of matrices to prove that the
TUM property of our constraint matrix is violated on addition of external constraints
for the MCF formulation:
1. A matrix with all elements in {0,+1,−1} is TUM if it contains no more than
one +1 and no more than −1 in each column.
We illustrate its proof using a regularly sampled 2D data set for ease of understanding.
In the case of the original MCF formulation (Costantini, 1998), each edge of the
unwrapping grid is traversed once in the clockwise and once in the anti-clockwise
direction. If we assume a reference direction, the column of the constraint matrix
corresponding to the flow variable Ki would have exactly one +1 and −1 in the
rows corresponding to the loops of which it is a part. The constraint matrix
satisfies property (1) shown above and hence is TUM. Suppose there were two
GPS measurements at pixels p and q. Then the additional external constraint,
corresponding to the MCF formulation, is of the form
q∑p
= ΔNp,q −q∑p
[ψi − ψj
2π
]
where (i, j) is an edge along any chosen path from point p and point q and ΔNp,q
is the expected number of cycles in unwrapped phase, as observed by GPS. Adding
this constraint violates property (1) above, and the resulting constraint matrix for
the MCF formulation is not TUM.
In case of the edgelist formulation, adding external constraints is equivalent to
adding additional edges to the unwrapping grid. Following arguments presented in
Section 5.5.1, the node incidence matrix for any directed grid is TUM. This implies
CHAPTER 5. PHASE UNWRAPPING 80
Figure 5.10: Map showing the location of the 40 km x 40 km area (red) being analyzedusing PS-InSAR. Fault traces for the Central San Andreas Fault and CalaverasFault are also shown in black. Locations of the Melendy Ranch and Slack Canyoncreepmeters are also shown.
that the linear programming relaxation still produces an exact solution for the edgelist
formulation including external constraints.
5.7 Case study: Creeping section of the San
Andreas Fault
We applied our edgelist unwrapping algorithm to a persistent scatterer InSAR (PS-
InSAR) data set covering an area of 40 km x 40 km around Monarch and Austin
peaks (Figure 5.10) in the Central San Andreas Fault region. Conventional InSAR
stacking in earlier studies characterized the spatial variation in slip deficit on the
Central San Andreas Fault (Ryder and Burgmann, 2008). However, the presence
of large decorrelated areas close to the fault severely compromised the ability to
reliably estimate the deformation just north of the fault (see Figure 2 from Ryder
and Burgmann (2008)).
We processed 21 SAR scenes (Track 27, Frame 2781) acquired by the ERS-1 and
ERS-2 satellites between 1992 and 2004. We selected a scene from March 1997 as
CHAPTER 5. PHASE UNWRAPPING 81
Figure 5.11: (Left) This figure shows the skeletal framework on which the PS-InSARdata set is unwrapped. Each slice represents an interferogram in the PS-InSAR time-series. All the node potentials in areas marked by red are set to zero. The plane ofthe fault trace is drawn in blue. (Right) The cost of the edges cutting across the faultis subsidized as a distance of center of edge from the fault as shown.
the master scene to minimize the combination of the perpendicular baseline and the
temporal baseline in order to optimize the accuracy of the correlated phase estimates
in the sparse PS network and generated 20 interferograms. We applied the maximum
likelihood persistent scatterer (MLPS) selection algorithm (Shanker and Zebker, 2007)
and found a sparse network of 2067 PS points per interferogram. Although the MLPS
algorithm identified more PS points than other public domain algorithms, the PS
density (1 per sq km) is still very low compared to the suggested threshold of 4
PS / sq km (Colesanti et al., 2003) recommended for conventional PS-InSAR phase
unwrapping algorithms.
Figure 5.12(a) shows the average LOS displacement rate in mm/yr and Fig-
ure 5.12(b) shows the average profile computed as a function of distance from the
fault as computed by our algorithm (Equations 5.11-5.14, 5.18-5.19 above). To form
this estimate, we also
1. Neglected elevation variation in the atmospheric phase screen, as the topography
does not change by more than a few hundred feet in this area. Thus we compute
unwrapped solution on a 3D grid similar to the Quasi-L∞ norm algorithm of
Hooper and Zebker (2007). This is different from the two-step approach of the
CHAPTER 5. PHASE UNWRAPPING 82
Figure 5.12: (Left) Average LOS displacement rate image estimated using the edgelistphase unwrapping algorithm. The fault trace and the area used for computing theaverage profile is also indicated. (Right) The average LOS displacement rate (reddots) as a function of distance from the fault is shown. Assuming all the displacementwas purely due to strike slip motion across the fault, we estimate a slip rate of 22mm /yr.
step-wise 3D algorithm (Hooper and Zebker, 2007) and the space-time MCF
algorithm (Pepe and Lanari, 2006).
2. Introduced an interferogram consisting purely of zero interferometric phase
values to represent the combination of the master scene with itself, into the
time-series. All the node potential variables (ni) for the vertices on the zero
interferogram were constrained to be zero (See Figure 5.11a). This establishes
a zero-reference frame with no discontinuities across the fault.
3. Chose the PS point with the highest temporal coherence as the master PS point
(see Figure 5.11a). The phase of pixels in all the interferograms were referenced
to the phase of the master PS point. The node potential for the master PS point
is also constrained to be zero in all the interferograms. Thus, all the unwrapped
phase values are estimated with reference to the master PS point.
4. Forced the solution to concentrate phase changes in the immediate vicinity of
the fault trace, by identifying a region of 1 km width along the active fault
CHAPTER 5. PHASE UNWRAPPING 83
trace in which all cost functions (Cij) associated with edges of the Delaunay
triangulation and all time-edges of the points that lie within this zone were
decreased. (See Figure 5.11b) This incorporates our knowledge that most of
the creep is near the surface and that deformation decreases with distance from
the fault. This is a good assumption for this fault but may not pertain to other
parts of the world.
One of the main advantages of our method is that we do not require a temporal
model for estimating deformation. The approach described above models the points
on opposite sides of the fault as connected by loose (subsidized cost) strings, i.e., the
unwrapped solution for a pixel depends more on the phase values of pixels on the
same side of the fault as itself and less on the phase of pixels on the other side of
the fault. Phase cycle jumps are preferentially compensated in the buffer zone. As a
consequence, the unwrapped solutions in the buffer zone may not be entirely reliable
due to the artificial discounts applied on the cost functions. But as we move away
from the buffer zone, the solutions should be more accurate. Figure 5.12 shows the
results obtained using our new algorithm.
It is useful to compare our solution to the results obtained by other algorithms.
We have already noted the comparison with the Ryder and Burgmann (2008) stack,
and see that the current solution produces estimates closer to the fault. We also
reduced the data using the step-wise 3D phase unwrapping algorithm (Figure 5.14)
(Hooper and Zebker, 2007). In this analysis we clearly see the presence of the fault,
but there appears “leakage” of signal across the known fault trace. This likely can
be attributed to the inability to accommodate multiple cycle jumps across the fault
for the large temporal baseline interferograms in the data set. We overcome this
problem by subsidizing the costs for edges cutting across the fault. In this area
the fault creeps at roughly 22 mm/yr (approximately 7 mm/yr in the radar line of
sight, LOS), so that interferograms with temporal separation of more than 4 years
exhibit multiple cycles across the fault. In an L1 or L2 norm formulation, multiple
cycle jumps are penalized more heavily than single cycle jumps and require careful,
CHAPTER 5. PHASE UNWRAPPING 84
and often impractical, adjustment of the cost functions. Also, the San Andreas
Fault fully bisects the image and forms a line discontinuity. In other words, there
is no direct connection between the regions on either side of the fault through an
area that is not noisy in phase. As the phase unwrapping problem is set up as a
minimiazation problem (Equations 5.11-5.14) multiple-cycle jumps across the fault
are underestimated. The edges cutting across the fault are incorrectly unwrapped in
the temporal unwrapping stage of the step-wise 3D algorithm (Hooper and Zebker,
2007), resulting in both overly smooth solutions and leakage of deformation signal
across the fault.
Figure 5.13 shows the cumulative LOS displacements between regions that are
located at a distance of 8 km on either side of the fault. For reference, the time-series
from the Melendy Ranch and Slack Canyon creepmeters (courtesy USGS) are also
included. The creep rates estimated by our method are consistent with the long-term
trend of observations from the USGS creepmeters in the area. However, the PS-
InSAR time-series exhibits a non-linear trend that needs to be further investigated.
This is seen in the Ryder and Burgmann (2008) InSAR stack as well. We note that
InSAR is more sensitive to the vertical component of the deformation than the lateral
component, due to the steep look angles of the instrument. GPS stations in the area
fail to characterize vertical deformation with comparable accuracy and are often not
used in modeling any vertical motion associated with creep across the fault. Hence,
vertical deformation could be one source of this observed non-linear trend that needs
further investigation with additional time-series InSAR studies using data from more
coherent high resolution X-band and L-band SAR sensors like TerraSAR-X and ALOS
PALSAR.
Most creep studies in the Central San Andreas Fault region use a constant velocity
to model the slip rate distribution at depth (Ryder and Burgmann, 2008; Rolandone
et al., 2007). Other studies, such as Nadeau and McEvilly (2004), suggest that
the recurrence of repeating microearthquakes affects the slip rate along the fault
significantly. Comparing results with Nadeau and McEvilly (2004), we also observe a
CHAPTER 5. PHASE UNWRAPPING 85
1992 1996 2000 2004−50
−30
−10
10
30
50
Ave
rage
LO
S di
spla
cem
ent i
n m
m
MLPSMelendy RanchSlack Canyon
Figure 5.13: The cumulative displacement time-series for the regions marked out inFigure 5.12(a). The time-series from the creep meters at Melendy ranch and Slackcanyon are also included for comparison. The creepmeter measurements have beenprojected on to the radar LOS direction for comparison.
CHAPTER 5. PHASE UNWRAPPING 86
Figure 5.14: The average LOS displacement rate in mm/yr estimated using thestepwise 3D unwrapping algorithm (Hooper and Zebker, 2007).
similar pulse-like variation in slip rate in the time period 1992-1996 in Figure 5.13. If
the results from our phase unwrapping method are plausible, the algorithm forms the
basis for a new method to monitor the changing strain field over the entire Central
San Andreas Fault area – spaceborne PS-InSAR is capable of measuring very small
displacements over a large area.
5.8 Discussion and Conclusions
We present a new phase unwrapping algorithm that is as accurate but more flexible
than previously known formulations. Our implementation exploits the TUM property
that allows us to solve large scale phase unwrapping problems using LP solvers. Tests
using simulated and real data sets demonstrate the validity of our new formulation.
We implemented the edgelist algorithm using the simplex modules of the CPLEX
software (CPLEX, 2006). In the absence of specialized constraints, illustrated in
CHAPTER 5. PHASE UNWRAPPING 87
the case study above, the edgelist formulation is computationally very inefficient
compared to the original MCF algorithm and requires more computer resources. The
quality of the solution however, matches that of the MCF formulation and permits a
simplified representation of multi-dimensional phase unwrapping problem and the
incorporation of data from external sources, like GPS, where available to better
constrain the unwrapped solution.
The edgelist method provides a way to apply a priori knowledge to improve our
ability to unwrap challenging data sets such as our case study above. Moreover,
the algorithm is easier to implement at a smaller scale with fewer number of
constraints and variables. This makes it an ideal candidate for improving the temporal
unwrapping step of the multi-temporal PS-InSAR and SBAS phase unwrapping
(Hooper and Zebker, 2007; Pepe and Lanari, 2006).
Chapter 6
Comparison of time-series InSAR
methods: Rockslide mapping in Norway
In this chapter, we apply the PS-InSAR methods developed in Chapter 4 and
Chapter 5 to study several rockslide sites in Troms County in the far north of Norway.
In the process, we also the compare the results from our PS-InSAR technique against
those from the Small Baseline Subset (SBAS) algorithm (Berardino et al., 2002).
We also take this opportunity to address the difference and similarities between the
PS-InSAR and the SBAS multitemporal InSAR methods for displacement studies in
rural terrain.
6.1 Introduction
The natural hazard of rockslides has a high socioeconomic and environmental
importance in many countries. Norway is particularly susceptible to large rockslides
due to its many fjords and steep mountains. One of the most dangerous hazards
related with rock slope failures are tsunamis that can lead to large loss of life. It
is therefore important to systematically identify potentially unstable rock slopes.
Traditional landslide monitoring techniques are expensive and time consuming.
Differential satellite interferometric synthetic aperture radar (InSAR) is an invaluable
tool for land displacement monitoring, particularly for the slow and very slow
88
CHAPTER 6. ROCKSLIDE MAPPING 89
landslides (See Hungr et al. (2001) for classification). Improved access to archived
satellite data has led to the development of several innovative multitemporal
algorithms, including Persistent Scatterer InSAR techniques. We also study the same
area with another popular time-series InSAR technique - the small baseline method
(SBAS).
Being a mountainous country, Norway is particularly susceptible to large rock
avalanches. In the last 100 years, over 170 people have been killed by tsunamis in
fjords caused by large rock avalanches. In each case, the rock avalanche was preceded
by many years of slow movement, with acceleration prior to slope failure (Eiken,
2008). With several thousand kilometers of inhabited coastline and valleys, it is a
challenge to identify the hazards efficiently. Even when we suspect an area to be
sliding, it may take several years of measurements to confirm it and extensive ground
instrumentation to characterize the type of motion.
The evolution of potential rockslides has often been studied using structural
geological methods (Braathen et al., 2004; Chigira, 1992; Agliardi et al., 2001).
Recent studies concentrate on documenting rockslide kinematics and identifying
geometric configurations particularly susceptible to sliding, e.g., (Braathen et al.,
2004; Henderson and Saintot, 2009). Less emphasis has been placed on the direct,
empirical relationships between the development of structures, evidence for movement
and subsequent effects on the geomorphological architecture (Colesanti andWasowski,
2006). Reconciliation of field observations and conventional measuring techniques
such as Global Navigation Satellite System (GNSS) receivers and Total Station
measurements has often proven ambiguous or problematic. Such quantification is
a necessary step in hazard and risk assessment (Solheim et al., 2005).
Recently, the potential of a differential synthetic aperture radar approach has
been investigated to study landslides by numerous research groups (Berardino et al.,
2003; Hilley et al., 2004; Strozzi et al., 2005; Rott and Nagler, 2006; Colesanti
and Wasowski, 2006). The interferometric phase measurements, however, are
affected by various decorrelation phenomena (Zebker and Villasenor, 1992). The
CHAPTER 6. ROCKSLIDE MAPPING 90
main limiting factors are atmospheric artifacts that can introduce a bias in the
phase measurement (Zebker et al., 1997), and temporal decorrelation that makes
InSAR phase measurements unreliable over vegetated regions due to the change in
relative position of the scatterers in a resolution element (Zebker and Villasenor,
1992). Another limitation is spatial baseline decorrelation which occurs when the
interferometric baseline is not exactly zero. Since the radar receives the coherent sum
of all independent scatterers within the resolution cell, these contributions are added
slightly differently due to the different geometry. Spatial decorrelation leaves many
image pair combinations infeasible in areas with steep terrain (Section 2.4).
Effects of various decorrelation phenomena can be reduced by combining multiple
SAR observations using multi-temporal InSAR techniques. Using more than two SAR
scenes leads to redundant measurements that can be used in more advanced time
series methods. These methods can be broadly classified into Persistent Scatterer
(PS-InSAR) and Small BAseline Subset (SBAS) methods. PS-InSAR methods work
by identifying image pixels in a stack of interferograms generated with the same
master that are coherent over long time intervals (Ferretti et al., 2000; Hooper et al.,
2004; Kampes, 2006). On the other hand, SBAS methods use all possible SAR image
combinations with a short spatial baseline to reduce the effects of spatial and temporal
decorrelation (Berardino et al., 2002; Lanari et al., 2007b).
Each of these methods has its advantages and limitations, and both have proven
to be effective in successfully estimating deformation time series in various regions
(Lanari et al., 2004a; Hilley et al., 2004; Hooper et al., 2007). In this chapter, we
compare the PS-InSAR and SBAS methods as applied to rockslide mapping.
This chapter is organized as follows. In Section 6.2 and Section 6.3 we present the
study area as well as the available SAR data. We also briefly discuss the geological
setting of our test area, including the relevance of PS-InSAR and SBAS methods
for analyzing such terrain. A short review of the SBAS implementation including
a discussion of theory, system issues and implementation is provided in Section 6.4.
In Section 6.5, we present the results obtained with both the PS-InSAR and SBAS
CHAPTER 6. ROCKSLIDE MAPPING 91
methods when applied to our test area. In Section 6.6 we discuss the results, and in
particular the ability of both the time-series InSAR methods to handle various error
sources in interferograms. Finally, Section 6.7 provides our conclusion and outlook
on these new developments in InSAR and their impacts on the field of remote sensing
of rockslides.
Figure 6.1: Photomontage illustrating typical scattering mechanisms that can beexpected from the rockslide sites. (a,b). Gamanjunni. (c) Nordnes.
6.2 Study area
Since 2003, a comprehensive study involving several different institutions and
international partners has focused on the investigation and documentation of the
possibility for large rockslides in Troms County. Aerial photo analysis, field mapping,
InSAR, LIDAR, differential GPS, continuous laser monitoring and 2D resistivity
CHAPTER 6. ROCKSLIDE MAPPING 92
analysis have been employed to identify potentially unstable rockslopes and further
focus on those that are actively moving. A total of 75 sites has been identified
(Henderson et al., 2009b) by means of aerial photo analysis, regional satellite InSAR
and helicopter reconnaissance, all with volumes ranging from 1 million m3 to 500
million m3. Geometric and kinematic models of movement of the different rockslides
have both been identified and placed within a regional geometric and tectonic model
where active faulting is both observed on the regional InSAR data and invoked as the
driving mechanism for the regional distribution of rockslides (Osmundsen et al., 2009).
Detailed studies linking geology, geomorphology and the InSAR techniques presented
here have been carried out on the scale of individual rockslides (Henderson et al.,
2009a). We focus on three rockslides (Gamanjunni, Rismmalcohkka and Nordnes),
located in Troms County in the far north of Norway.
The photomontage in Figure 6.1(a–c) illustrates typical terrains in our test area.
PS-InSAR methods rely on identification of pixels characterized by a dominant
scatterer and operate on interferograms of the highest possible resolution. SBAS
methods, on the other hand, are designed to identify pixels characterized by a
distributed scattering mechanism and hence, operate on multi-looked interferograms
(Section 2.5). In our area of interest, we can expect to observe a mixture of all types.
We therefore expect that PS and SBAS methods will be complementary.
6.3 Available SAR data
Our analysis is based on 18 unique European Space Agency (ESA) ERS-1 and ERS-
2 SAR data acquired from 1992 through 1999 (descending orbit, track 251, frame
2196), see Table 6.1. We selected only snow-free scenes. The ERS satellites have an
operating wavelength of 5.66 cm, and the radar looks to the right (west for descending
imaging geometry) with an angle of approximately 23.5◦ from the vertical. The radar
is only sensitive to displacement changes with a component in the radar line-of-sight
(LOS) direction.
CHAPTER 6. ROCKSLIDE MAPPING 93
Figure 6.2: Landsat location map of study area in Troms County showing meanLOS velocity for the period of study (SBAS). Most of the identified rockslide sitesare clustered around the presently active (Osmundsen et al., 2009) Lyngen FaultComplex. Location of study area in Troms County in Norway shown in inset. Theanalyzed rockslides are marked with white rectangles.
Figure 6.3a shows the 65 interferograms that were computed using a maximum
spatial baseline threshold of 600 m for the spatial baseline and a 5 years for the
temporal baseline. Each line corresponds to an interferogram between two acquisition
dates. We process the interferograms using the Norut GSAR software (Larsen et al.,
2005). We used a digital elevation model from the Norwegian Mapping Authority
with a grid size of 25× 25 m and a height standard deviation of 5–6 m to remove the
topographic phase. A complex multi-look operation taking two looks in range and
CHAPTER 6. ROCKSLIDE MAPPING 94
Figure 6.3: Baseline plot. (a) SBAS baselines relative to 1995-06-01. (b) PS baselinesrelative to 1995-06-02.
eight looks in azimuth, produced pixels with ground range dimensions of about 50 m
in range and 50 m in azimuth direction.
Figure 6.3b shows the 15 interferograms computed with respect to the selected
master scene (1995-06-02) for PS analysis. All interferograms were computed at full
resolution, with pixel dimensions of approximately 20 m in ground range and 4 m in
azimuth. The interferograms also span the time period from June 1992 to September
1999. We used the Norut GSAR software (Larsen et al., 2005) for generating all
interferograms. The topographic phase component was corrected using the same
DEM as the one used for the SBAS analysis. We estimated topography related
atmospheric phase screen for each interferogram and compensated before PS-InSAR
analysis, as is described below in Section 6.3.1.
6.3.1 Atmospheric phase correlated with topography
The atmospheric phase component correlated with topography is a significant noise
source in interferograms, especially in areas with varied topography (Onn and
Zebker, 2006). The neutral atmospheric phase screen often is linearly dependent
on topography as seen in GPS and InSAR data (Emardson et al., 2003). Onn
and Zebker (2006) showed that the topography-correlated neutral atmospheric
phase screen can be empirically corrected using the phase information from the
CHAPTER 6. ROCKSLIDE MAPPING 95
interferograms themselves. Figure 6.4 shows a scatter plot of the interferometric phase
for an example interferogram from Southern California against the DEM height (Onn
and Zebker, 2006). A fit to these data provides the correction function.
Figure 6.4: (Top) Interferometric phase (blue) and the GPS Zenith Wet Delay (red)plotted as a function of DEM height for an example interferogram in SouthernCalifornia. For details see Onn and Zebker (2006). (Bottom) Interferometric phaseof all pixels with coherence over 0.6 plotted as a function of height for an exampleinterferogram over our test area.
A module to automatically estimate the phase screen as a linear function of the
DEM height was incorporated into the GSAR processing chain following Emardson et
al. (2003). The topography-related atmospheric phase component was compensated
in all the interferograms used for producing results in Section 6.5 before the actual
time-series processing.
CHAPTER 6. ROCKSLIDE MAPPING 96
Table 6.1: Used scenes from track 251, frame 2196.
Mission Date B⊥a Bta Number of SBAS interfero-
grams that scene is part of
(yyyy-mm-dd) (m) (days)
ERS-1 1992-06-23b -696 -1074 5
ERS-1 1992-07-28 -120 -1039 10
ERS-1 1992-09-01 310 -1004 10
ERS-1 1993-07-13 -491 -689 7
ERS-1 1993-09-21 647 -619 4
ERS-1 1995-06-01d 31 -1 10
ERS-2 1995-06-02ce 0 0 0
ERS-1 1995-07-06 -150 34 11
ERS-2 1995-07-07c -154 35 0
ERS-1 1995-09-14 -182 104 11
ERS-2 1995-09-15 -123 105 11
ERS-1 1995-10-19 644 139 6
ERS-2 1995-10-20 706 140 5
ERS-2 1997-06-06 -14 735 8
ERS-2 1997-08-15 368 805 12
ERS-2 1997-09-19 -213 840 10
ERS-2 1999-08-20b 1175 1540 2
ERS-2 1999-09-24 78 1575 8
a Baseline relative to the scene 1995-06-02.b Only in SBAS.c Only in PS.d SBAS master geometry.e PS master geometry.
CHAPTER 6. ROCKSLIDE MAPPING 97
6.4 Short description of the SBAS algorithm
In this section, we review the fundamental concepts behind the SBAS multitemporal
InSAR algorithm. For further details, we refer the readers to Berardino et al. (2002),
Casu et al. (2006) and Lanari et al. (2007b).
Let us start by assuming a set of N+1 coregistered single look complex (SLC) SAR
images. The first step of the SBAS algorithm is the generation ofM multilooked small
baseline differential interferograms. By restricting the maximum spatial baseline used,
spatial decorrelation is reduced, and the effect of residual phase due to uncompensated
topography is mitigated. On the other hand, such a baseline constraint can lead to
a separation of the interferograms into several independent subsets in the baseline-
time domain. The SBAS algorithm correctly retrieves the phase values as long as the
subsets are overlapping in time.
We select a common set of coherent pixels in M interferograms as characterized
by a high coherence in a selected fraction of the interferograms.
The SBAS algorithm relies on the availability of unwrapped phase values for
all M interferograms (Berardino et al., 2002). Consequently, a phase unwrapping
step is needed to retrieve the absolute phase values from the (modulo 2π) wrapped
interferometric phase (Goldstein et al., 1988; Ghiglia and Pritt, 1998; Costantini,
1998; Chen and Zebker, 2001). A best fit plane is removed from the unwrapped
phase to account for possible precise orbit errors in the unwrapped data.
After phase unwrapping and ramp removal, all pixels are referenced to a selected
point, usually characterized by high coherence and a priori known deformation
temporal model. The differential phase for a generic coherent pixel, in interferogram
j, generated by combining SAR acquisitions at times tB and tA is (Berardino et al.,
CHAPTER 6. ROCKSLIDE MAPPING 98
2002)
δφj = φ(tB)− φ(tA)
≈ 4π
λ[d(tB)− d(tA)] +
4π
λ
B⊥j
r sin θΔz
+4π
λ[datm(tB)− datm(tA)]
+ nj, ∀j = 1, ...,M,
(6.1)
where λ is the transmitted radar wavelength, φ(tB) and φ(tA) are the phases
corresponding with the times tB and tA, d(tB) and d(tA) are the radar LOS projection
of the cumulative deformation referenced to the first scene (i.e. implying φ(t0) = 0).
We also include a phase term that is related to possible errors Δz in the applied
digital elevation model (DEM) used for differential interferogram generation. This
phase component is proportional to the perpendicular baseline for each interferogram
B⊥j, range distance r, as well as satellite look angle θ. A possible atmospheric signal
(Zebker et al., 1997; Ferretti et al., 2000; Hanssen, 2001) is included in the terms
datm(tB) and datm(tA). Decorrelation effects and other noise sources are included
in the last term nj. Note that the following processing steps are carried out on a
pixel-by-pixel basis on all selected coherent pixels.
6.4.1 Estimation of DEM error and low pass displacement
signal
We first estimate DEM error and displacement model parameters. We use a cubic
temporal deformation model
φ(ti) = v · (ti − t0) +1
2a · (ti − t0)
2 +1
6Δa · (ti − t0)
3
=4π
λ· d(ti) (6.2)
CHAPTER 6. ROCKSLIDE MAPPING 99
and write the unknown parameter vector as
xT = [v, a,Δa,Δz], (6.3)
where v is mean velocity, a is mean acceleration, Δa is mean acceleration variation,
and Δz is DEM error.
As in (Berardino et al., 2002), using a cubic model, we form a linear system with
M equations (corresponding to the vector of unwrapped phase values δφ) and 4
unknowns
[AM, c]x = δφ, (6.4)
whereM is anN×4 matrix corresponding to the parameters of the cubic displacement
model in Equation 6.2, cT = [(4π/λ)(B⊥1/r1 sin θ1, ..., (4π/λ)(B⊥M/rM sin θM ] is a
factor proportional to the DEM error, and A is an M ×N difference operator matrix
corresponding to the unknown cumulative phase values φ = [φ1, φ2, ..., φN ] and the
M interferograms. For details about A and M, see (Berardino et al., 2002).
We would like to point out that the cubic displacement model in Equation 6.2 is
used only to help the estimation of the DEM error, while the final inversion to compute
the displacement time series is independent of such a model, see Equation 6.5.
Since we apply a displacement model, the equation system in Equation 6.4 is
generally simplified. For such a smooth temporal model, the product [AM, c] is
nonsingular. The estimate of x can then be obtained in an optimal least squares (LS)
way (Golub and Loan, 1996).
6.4.2 Estimation of cumulative phase
The estimated DEM error is subtracted from the unwrapped phase, forming the
following linear system
Bv = δφ′, (6.5)
CHAPTER 6. ROCKSLIDE MAPPING 100
where δφ′ = φ−Δz · c is the redundant set of phase differences with the DEM error
contribution subtracted, and the unknown coefficient vector is
vT =
[v1 =
φ(t1)
t1 − t0, ..., vN =
φ(tN)− φ(tN−1)
tN − tN−1
]. (6.6)
As in (Berardino et al., 2002), we replace the unknown cumulative phase values
φT = [φ1, φ2, ..., φN ] with the mean phase velocity between time adjacent acquisitions.
The design matrix B is changed accordingly, and now represents the cumulative time
between each interferogram pair. Accordingly, a trivial integration step is needed to
recover the final solution φ.
6.4.3 Atmospheric filtering and final displacement time se-
ries estimation
Atmospheric inhomogeneities can create an artificial atmospheric signal superimposed
on each SAR image (Zebker et al., 1997; Hanssen, 1998; Hanssen, 2001).
The atmospheric filtering approach used in the SBAS method is inspired by
Ferretti et al. (2000) and Ferretti et al. (2001). The rationale behind the SBAS
algorithm is to separate the different phase contributions (deformation, topographic
error, atmospheric signal and decorrelation noise) by inverting a linear system of
equations. In the atmospheric filtering step we exploit the spatial and temporal
correlations of the atmosphere. The key observation is that the atmospheric phase
component is highly correlated in space, but uncorrelated in time (Hanssen, 1998;
Hanssen, 2001). The undesired atmospheric signal is estimated as follows:
1. the low pass deformation signal from Equation 6.2 is removed from the estimated
deformation time series.
2. Following this, the residual phase signal is detected as the result of first, a
temporal high pass filter with respect to the time variable, then a spatial low
pass filter on the residuals.
CHAPTER 6. ROCKSLIDE MAPPING 101
Once the atmospheric signal has been estimated, it is subtracted from the estimated
deformation phase signal. To obtain the final displacement estimate, the phase must
be multiplied by the factor λ/4π.
6.5 Results
In this section we present the results from using the two multi-temporal InSAR
techniques, PS-InSAR and SBAS. We focus on three specific rockslides where
displacement has been identified and confirmed by field surveys performed by the
Geological Survey of Norway (NGU) (Henderson et al., 2009a; Henderson et al.,
2009b).
Common processing for both SBAS and PS-InSAR includes SAR processing and
coregistration. We focused all raw SAR scenes into SLC images using the Norut
GSAR processor (Larsen et al., 2005). After focusing, we prewhitened and equalized
the Fourier spectra of all SLCs, and coregistered the SLC images to a selected master
geometry that was chosen to minimize both spatial and temporal baseline dispersion.
6.5.1 PS-InSAR procedure
For each complex interferogram, we first compute an amplitude calibration constant
by summing up the absolute values of the area of interest (Lyons and Sandwell,
2003). We select an initial set of PS candidates after normalization of the amplitudes
using the calibration values and a low threshold on the amplitude dispersion index
(Ferretti et al., 2001). We estimate phase contributions due to DEM errors using
the perpendicular baseline estimates for each pixel and then subtract them from
the measured interferometric phase values. The spatially correlated phase terms are
iteratively determined for each of these PS candidates using the StaMPS software.
We then apply the MLPS selection algorithm with a ML correlation threshold of
0.7. The identified PS network is then used to re-estimate all the spatially correlated
CHAPTER 6. ROCKSLIDE MAPPING 102
phase terms for each PS pixel. No explicit orbital ramp removal is necessary as this
slowly varying term also includes the spatially correlated phase term.
We used the stepwise-3D phase unwrapping algorithm (Hooper and Zebker, 2007)
to unwrap the sparse phase measurements. No deformation model was applied to
aid the unwrapping process. Atmospheric phase screen and orbital phase ramps were
estimated using a combination of a low-pass spatial filter and a high-pass temporal
filter. A detailed analysis of the estimated deformation time series for three selected
rockslides is presented in Section 6.6.
6.5.2 SBAS procedure
For each complex interferogram, we estimated and removed a best fitting linear phase
ramp due to imprecise orbit knowledge (Lauknes et al., 2005). After removal of the
orbital phase ramp, the differential phase delay due to tropospheric stratification is
estimated for each interferogram (Section 6.3.1). The main principle can be outlined
as follows. Based on the individual differential phase estimates (two dimensional
interferograms), we form a linear system where we invert the phase as a function of
height for each SAR scene using a least squares approach. The resulting differential
phase contribution due to atmosphere is then removed from each input interferogram.
In order to exclude decorrelated areas from the study, and to make phase
unwrapping feasible, we applied a pixel thresholding, selecting only the pixels
that exhibit an estimated coherence value larger than 0.25 in at least 30% of the
interferograms. The overall phase coherence of the area is relatively high due to the
limited vegetation cover above 600–700 m elevation.
Based on the selected pixels we apply a Delaunay triangulation and interpolation
on all images (Costantini and Rosen, 1999). Thus, we are able to link spatially sep-
arated coherent patches. Following this operation, we unwrapped the interferograms
using the SNAPHU software (Chen and Zebker, 2001). It should be noted that our
preprocessing steps (orbital error ramp and stratified atmospheric removal) are crucial
for a successful spatial unwrapping in this terrain.
CHAPTER 6. ROCKSLIDE MAPPING 103
After removing orbital trends and topography related atmosphere, as well as
replacing the phase of low coherence points with Delaunay triangulated phase, we
are left with a spatially correlated atmospheric signal. After the phase unwrapping
step, all pixels are referenced to a chosen point on the eastern shore of the Lyngen
fjord.
We then applied the SBAS algorithm described in Section 6.4 to the 65 InSAR
pairs. For phase retrieval, we applied the new L1 norm based SBAS phase inversion
method presented in (Lauknes et al., 2009).
The spatially correlated atmospheric contributions were estimated and filtered out
before estimating a mean displacement velocity. Figure 6.2 shows the mean velocity
rate over the analyzed area.
6.5.3 Differences in implementation
We summarize the main differences between the PS-InSAR and SBAS implementa-
tions in Table 6.2. Fundamentally, there are three major differences between our
PS-InSAR and SBAS implementations:
1. The PS-InSAR method does not require any predetermined displacement
model and yet successfully manages to identify the expected trend in LOS
displacement. The SBAS method needs a temporal displacement model to
separate DEM error from displacement.
2. We used the stepwise-3D algorithm (Hooper and Zebker, 2007) to unwrap the
sparse 3D data set in our PS-InSAR implementation, while two dimensional
phase unwrapping using the SNAPHU algorithm (Chen and Zebker, 2001)
sufficed for our SBAS implementation.
3. Spatial filtering is an essential step in the StaMPS framework for identification of
the PS pixels. No spatial filtering is needed in the SBAS inversion. Deformation
estimates on a pixel-by-pixel basis results from inverting the unwrapped phase.
CHAPTER 6. ROCKSLIDE MAPPING 104
Figure 6.5: Detailed results from the Gamanjunni slide (see area G, Figure 6.2). (a–b)show the estimated mean velocity displacement rate maps overlaid on an aerial photo,while (c–d) show the same results draped on top of an aerial photo, for SBAS (a,c)and MLPS (b,d). (e–f) show the estimated time series for both SBAS and MLPS forpoints A and B, respectively. The mapped active structures are marked in (a–b).
6.5.4 Final displacement estimates
Figures 6.5–6.7(a–d) show the estimated mean LOS velocity in (mm/year) using both
SBAS and MLPS methods. The mean velocity has been draped on top of an aerial
photo in (a,b), and a DEM has been used to produce the three dimensional perspective
view in (c,d).
CHAPTER 6. ROCKSLIDE MAPPING 105
Figure 6.6: Detailed results from the Rismmalcohkka slide (see area R, Figure 6.2).(a–b) show the estimated mean velocity displacement rate maps overlaid on an aerialphoto, while (c–d) show the same results draped on top of an areal photo, for SBAS(a,c) and MLPS (b,d). (e–f) show the estimated time series for both SBAS and MLPSfor points C and D, respectively.
It should be remarked that, in order to ease comparison with the PS results, we
have chosen to geocode each pixel from the SBAS results as an individual point.
Consequently, when plotting the points, it appears that the data are rather sparse.
Often, in reality, we do have complete spatial coverage.
CHAPTER 6. ROCKSLIDE MAPPING 106
Figure 6.7: Detailed results from the Nordnes slide (see area N, Figure 6.2). (a–b)show the estimated mean velocity displacement rate maps overlaid on an aerial photo,while (c–d) show the same results draped on top of an areal photo, for SBAS (a,c)and MLPS (b,d). (e–f) show the estimated time series for both SBAS and MLPS forpoints E and F, respectively.
Figures 6.5–6.7(e,f) show the estimated LOS displacement time series for two
selected points using both SBAS and MLPS methods. The positions of the two
selected point areas are indicated in the mean velocity maps in Figures 6.5–6.7(a–
d). The SBAS estimates have a pixel dimension of about 30× 40 m, and the MLPS
estimates retrieved at full resolution have a pixel dimension of 4×20 m, in the azimuth
and range directions, respectively. In order to compare time series, we selected an area
CHAPTER 6. ROCKSLIDE MAPPING 107
of 3×3 pixels from the SBAS (corresponding to about 90×120 m on the ground), and
used all PS pixels within this area. All points within the selected area were averaged
as to produce a better estimate. In order to produce the time series, we referenced
all points to a chosen reference area, marked with a star in the corresponding mean
velocity plots from Figures. 6.5–6.7(a–d).
The triangles in the time series (Figs. 6.5–6.7(e,f)) correspond to the SBAS
estimates, and the crosses correspond to the MLPS estimates. The lines correspond
to fitting a linear model fit to the SBAS (solid line) and MLPS (dashed line) points
using a least absolute deviation (L1) method.
In Table 6.3 we summarize the estimated mean velocities for the time series plotted
in Figures 6.5–6.7(e,f). In Table 6.3 we also show the computed root mean squared
error (RMSE) between the linear model and the data, for both SBAS and MLPS.
6.6 Discussion
For all three areas, the SBAS results have greater spatial coverage compared to
PS-InSAR. The absence of vegetation above 600–700 m altitude could explain this.
Perhaps in these areas, distributed scattering mechanism dominates over the point
target scattering mechanism. PS pixels possibly correspond to areas having large
boulders located inside the resolution cell. We expect that more PS points are
detected in areas where the scattering is of a “complex character”, i.e. an area with
mixed point scatterers and distributed scatterers, while the SBAS is better suited
for areas where the scattering mechanism is distributed (see Figure 6.1 for scattering
mechanisms). Our results show that the SBAS has greater coverage in all areas. This
is in agreement with the photos in Figure 6.1(a–b), where a dominating distributed
scattering is likely.
A look at the time series in Figures 6.5–6.7(e,f) shows that displacement estimates
corresponding to certain acquisitions tend to be noisier than the others. In SBAS,
these can probably be attributed to the number of interferogram combinations each
CHAPTER 6. ROCKSLIDE MAPPING 108
SAR scene contributes to, see Figures 6.3a and Table 6.1. If few interferograms
are computed with respect to a scene, we expect that the inversion will be less
robust for this point, and the noise level will be higher. In PS-InSAR, the noise
on the temporal time series can be attributed to the baseline of the corresponding
interferogram. In the StaMPS framework, spatially correlated terms are estimated
by filtering each interferogram. In the case of interferograms with large baselines and
sparse PS networks, these filtering estimates tend to be noisier.
It should be noted that PS-InSAR methods estimate the LOS velocity of the
dominant scatterer in a resolution element, whereas the SBAS method estimates the
average motion of all scatterers within a resolution cell. Therefore, slight variations
between the velocities might occur.
For our study area, which is snow covered during large parts of the year, we can
only use the summer and autumn scenes for analysis. Consequently, ascertaining
seasonal variation in the rockslide displacement rates is difficult.
Another effect of using only data from the summer-autumn period is that the data
points are clustered along the time dimension with a nearly periodic time spacing of
one year. This makes the temporal atmospheric filtering challenging since we need
to use long filter lengths. Furthermore, the irregularity of the temporal sampling can
lead to filter end effects.
The difference in the PS and SBAS spatial pixel density for an area also affects
our estimates of the two dimensional atmospheric signal, and hence the estimated
deformation series.
In this work, we compare our time series results at the site level. The SBAS
dataset has a better spatial coverage than the PS-InSAR dataset in this case. From
the time series in Figures 6.5–6.7(e,f) we can see that the two methods follow each
other quite well for all points selected. In general, the SBAS time series exhibit a
larger spread around the fitted linear curve than the MLPS points. On average, the
estimated root mean squared error (RMSE) between the linear fit and the data points
CHAPTER 6. ROCKSLIDE MAPPING 109
is 1.6 mm higher for SBAS than for MLPS, and the average mean velocity rates are
within 0.5 mm/year of each other (Table 6.3).
6.6.1 Gamanjunni
The Gamanjunni rockslide (Figure 6.5) is located on a west-facing mountain at 1200
metres elevation and is bounded by two back-scarps (Henderson et al., 2009a). The
block volume is approximately 17 million m3, and it is therefore among the biggest
potential rockslides in Norway.
On the outcrop scale, the InSAR results are in close agreement with field
observations. The spatial extent of the InSAR movement area matches extremely
well with the area delimited by the active structures (See Figure 6.5. The block is
subsiding at a rate up to 5 mm/year relative to the surrounding mountainside. Field
evidence suggests that some fault segments have been active at different times and
that previously active fault segments, which are now extinct, have been superseded
by younger, more active faults, which are accommodating movement at present.
The NNW-SSE trending internal structures in the failing block are reflected in a
NNW-SSE trending pattern in the InSAR data. This is seen in both the SBAS and
PS data. Henderson et al. (2009a) demonstrated an apparent upward movement in
the front of the block that is related to the failure mechanism, which is also faithfully
recorded in both data sets, although it is more apparent in the SBAS data.
As indicated by Henderson et al. (2009a), we clearly observe that we are able to
see differential movement patterns within the rockslide that can be directly correlated
to the displacement profile of the individual faults and to the variable surface
expression of the different faults. Based on the InSAR results, GPS monuments have
been established. The monuments are measured once a year, and the preliminary
differential GPS results are in broad agreement with the InSAR data (Eiken, 2008).
CHAPTER 6. ROCKSLIDE MAPPING 110
6.6.2 Rismmalcohkka
The Rismmalcohkka rockslide (Figure 6.6) is located on a south-west facing mountain
between 600–1000 m elevation. The block volume is approximately 100 million m3.
This rockslide geometry and architecture is fundamentally different from the sites
of Nordnes and Gamanjunni, which form a more blocky, contiguous rock mass. The
Rismmalcohkka slide covers a large area, but represents a rather “thin-skinned”
rockslide where the total depth to the sole thrust is no more than 50–80 metres. The
surface morphology consists of an extremely attenuated, rubbleized, moving mass.
Hence, it is plausible that a distributed scattering model of the SBAS technique is
better suited for InSAR analysis in this area compared to the dominant scatterer
model of PS-InSAR.
6.6.3 Nordnes
Nordnes rockslide (Figure 6.7) shows the biggest difference between the two multi-
temporal techniques among the areas analyzed. Both methods successfully identify
the large deformation signal on the higher part of the rockslide. The spatial coverage
of the PS-InSAR on the lower section is significantly lower than that of the SBAS
technique, possibly indicating the nature of the scattering mechanism.
It can also be noted that in the Nordnes site, the SBAS method is able to resolve
the faster moving, and more chaotically moving frontal part of the slide much better
than the PS-InSAR method. This is probably related to the fact that the failure
mechanism is much more chaotic in the most frontal part of the rockslide, with blocks
tumbling in different directions.
6.7 Conclusions
SBAS techniques have a great advantage in that they can easily recover regional
scale deformation, while PS techniques can be used for detailed and complimentary
studies. Furthermore, the convergence between structural and InSAR descriptions of
CHAPTER 6. ROCKSLIDE MAPPING 111
the active rockslides is remarkable. Ground observations of fault geometry, evolution
and developing footwall profile are precisely reflected in the SBAS and PS InSAR
results. InSAR data have contributed to the detailed interpretation of the failure
mechanism of several rockslides.
On the regional scale, large areas can be determined to be stable, thereby allowing
rather expensive field activities to be focused on areas where movement has been
detected. Our extensive experience in the ground-truthing of the InSAR method
demonstrates a remarkable correlation between subsidence detected from the InSAR
and mappable, unstable slopes in the terrain.
The millimetric precision of velocity determination, combined with the relatively
high spatial resolution, allows detailed site characterization of many large rockslides.
In addition to determining if a block is moving, we can discern subtleties such as
differential velocities along an actively developing fault scarp. Although InSAR
cannot be considered a replacement for ground based monitoring of active rockslides,
it provides an important complementary dataset.
InSAR provides a new technique to determine potential rockslide movement and
therefore provides a direct link between quantitative ground movement data and
the structures, kinematics and changes of slope. Integrated studies using geological,
geomorphological and newly developed remote sensing techniques are relatively few,
e.g., (Jaboyedoff et al., 2004). The direct linkage of InSAR data to structural geology
and sliding processes is a new and novel niche in rockslide research and will prove
to be a critical tool in future rockslide mitigation. Time-series InSAR techniques
provide a detailed velocity maps of large areas covering entire landslides as opposed
to sparse points on in case of GPS networks. This important characteristic of time-
series InSAR will see it play a more important role in flow modeling of landslides in
the future.
CHAPTER 6. ROCKSLIDE MAPPING 112
Table 6.2: Main differences between SBAS and PS.
SBAS PS
Main algorithms (Berardino et al., 2002;
Casu et al., 2006; Lanari
et al., 2007b).
(Ferretti et al., 2000; Fer-
retti et al., 2001; Werner
et al., 2003; Hooper
et al., 2004; Kampes,
2006; Shanker and Ze-
bker, 2007).
Interferogram generation Multilooked
interferograms using
the baseline thresholds as
discussed in Section 6.4.
Full resolution interfero-
grams with respect to the
chosen geometry master
image.
Pixel selection Based on estimated coher-
ence.
Based on estimated ML
correlation.
Target scattering mecha-
nism
Distributed scatterer
pixel.
Pixel with dominant scat-
terer.
Resolution Multi-looked (30× 40 m). Single look (4× 20 m).
Phase unwrapping Delaunay triangulation
and SNAPHU
(Costantini, 1998).
Stepwise-3D (Hooper and
Zebker, 2007).
Low pass filter Gaussian with dimension
1500 m.
Butterworth filter with
cutoff of 800 m.
Temporal filter Triangular (Bartlett) 400
days.
1 year Gaussian filter.
CHAPTER 6. ROCKSLIDE MAPPING 113
Table 6.3: Time series results from the three areas.
Area SBAS mean PS mean SBAS RMSE PS RMSE
(mm/year) (mm/year) (mm) (mm)
Gamanjunni
point A -3.80 -4.12 3.17 2.00
point B -5.57 -4.30 3.27 1.54
Rismmalcohkka
point C -4.16 -4.08 4.43 3.92
point D -3.34 -3.52 4.92 1.80
Nordnes
point E -0.10 -0.28 1.06 0.98
point F -4.08 -2.97 3.92 1.80
Chapter 7
Noise characteristics of PS-InSAR
time-series: A case study of the San
Francisco Bay Area
In this chapter, we describe the observed noise characteristics of deformation time-
series estimated using the MLPS algorithm (Shanker and Zebker, 2007) in the StaMPS
framework (Hooper, 2006). We use a dataset of 43 SAR scenes covering the San
Francisco Bay Area for our analysis. In particular, we focus on the (1) Hayward
Fault in the Oakland-Alameda region, (2) peninsula segment of the San Andreas Fault
near the San Francisco airport (SFO) and (3) South Bay region, all of them imaged
by ERS sensors during 1995-2000. First, we quantify the ability of the StaMPS
framework to correct temporally uncorrelated phase error terms by analyzing the
estimated deformation between ERS tandem scenes (separation of one day). We then
quantitatively compare and cross-validate our time-series PS-InSAR results againt
the results from the Small Baseline Subset (SBAS) InSAR time-series algorithm
(Berardino et al., 2002). We also compare our results against creep measurements
from alignment arrays along the Hayward Fault (McFarland et al., 2009).
We present in detail the estimated line of sight (LOS) velocities and deformation
time-series for selected stations using both the time-series techniques, and successfully
identify the salient deformation features in all three test regions. We show that the
114
CHAPTER 7. NOISE IN PS-INSAR 115
root-mean squared difference in estimate line of sight (LOS) deformation between
tandem pairs is about 4 mm. We also show that the root mean squared differences of
the estimated mean velocities and deformation between the PS and SBAS time-series
InSAR techniques are about 1 mm/yr and 5 mm, respectively. We also compare
our deformation estimates against those from the creep meters of the San Francisco
State University (SFSU) Fault Creep Monitoring Project. The observed discrepancies
are within expected noise levels and design parameters of the time-series InSAR
techniques.
7.1 Data
In this study, we compare deformation time-series processed over three sub-areas
in the San Francisco Bay Area (Figure 7.1) using two different multi-temporal
algorithms: Stanford University’s MLPS algorithm in the StaMPS framework (PS)
and IREA-CNR’s SBAS technique relevant to the SBAS family of algorithms.
We analyzed a total of 43 SAR acquisitions acquired during descending passes
of ERS-1 and ERS-2 satellites (Track 70, Frame 2853) between May 1995 and
December 2000. All the time-series InSAR products were produced using a common
geometry corresponding to an acquisition from December 1997. 42 common-master
interferograms were used for the PS analysis, and 124 multi-looked interferograms (20
azimuth looks and 4 range looks), corresponding to the network shown in Figure 7.2,
were used in the SBAS analysis. There are no significant time gaps in the data
set. Note that this area has been previously analyzed independently using these two
techniques (Shanker and Zebker, 2007; Lanari et al., 2007a; Lanari et al., 2007b)
but with different sets of SAR acquisitions. The SAR images were processed from
Level-0 data at both Stanford and IREA-CNR, and the results were produced in radar
coordinates for ease in direct pixel-to-pixel comparisons. The integer pixel offsets were
then computed using the master SAR scenes and the results were aligned at the pixel
level. The results from PS analysis were averaged over a window of 20 azimuth and 4
CHAPTER 7. NOISE IN PS-INSAR 116
Figure 7.1: The locations of three sub-areas that were used in the time-seriescomparison, shown overlaid on the SRTM DEM of the region. The largest squarerepresents the total area covered by the frame that was analyzed using the SBASmethod.
range pixels to match the resolution of the SBAS analysis results. The PS results were
processed as three different areal subsets due to the large data volume. The SBAS
results were processed for the complete frame see Figure 7.1. The PS data set was
unwrapped using the stepwise-3D phase unwrapping algorithm (Hooper and Zebker,
2007), while the SBAS time-series was estimated using the interferogram network
shown in Figure 7.2 and the extended MCF phase unwrapping algorithm (Pepe and
Lanari, 2006). The comparison of the results requires a common reference point. In
our case, we choose the common reference pixel as the one with the maximum average
temporal coherence (see Hooper (2006) and Pepe & Lanari (2006) for definition) as
estimated using the PS and SBAS techniques. We also use a temporal filter of 400
days length to mitigate the effects of the seasonal variation of the ground water levels.
These effects are significant in areas close to the bay (Lanari et al., 2007b).
Unfortunately, there were no continuous GPS stations operating in these areas
prior to 1999, and there were only two GPS stations with any overlap with our
CHAPTER 7. NOISE IN PS-INSAR 117
Figure 7.2: (Left) Time-baseline plot of the 43 ERS scenes (vertices) used for the PS-InSAR analysis. SAR scene from Dec 1997(square) was used for the master geometry.(Right) Edges between vertices represent the 124 interferograms used in the SBASanalysis. Again, the SAR scene from Dec 1997 (square) was used for the mastergeometry.
estimated time-series. Hence, we could not compare our deformation estimates
against those from continuous GPS stations from networks that have since been added
in the San Francisco Bay Area. San Francisco State University (SFSU) and the United
States Geological Survey (USGS) maintain a network of alignment arrays across the
active sections of the Hayward Fault (McFarland et al., 2009). Eight stations between
Kilometers 23 and 46 are covered by one the sub-areas that we analyzed and were
used for direct comparison with our estimated deformation estimates. The data is
freely available for download at http://pubs.usgs.gov/of/2009/1119/.
7.2 Methodology
We use three different methods to analyze our estimated deformation time-series.
In the first technique (Section 7.2.1), we focus on tandem pairs in the time-series.
No external geodetic observations are used to quantify noise levels in this case. In
the second technique, we compare our results against those obtained from another
indepedent time-series technique belonging to the SBAS family of algorithms in
Section 7.2.2. This comparison yields a measure of degree of consistency in the
estimated deformation signal by the two time-series InSAR techniques. Finally, we
CHAPTER 7. NOISE IN PS-INSAR 118
compare our results against ground truth in the form of creep measurements from
the SFSU-USGS alignment array.
7.2.1 Tandem SAR scenes
Let P (x, t) represent the estimated PS-InSAR deformation time-series with spatial
index “x” at time “t”. We identified three pairs of ERS tandem scenes (Duchossois
et al., 1996) in our set of 43 SAR scenes - 10 − 11 Nov 1995, 29 − 30 Mar 1996 and
03 − 04 May 1996 and analyzed the difference in estimated time-series on a pixel-
by-pixel basis. The deformation phase term (φdefo,x,t in Equation 3.14) does not
change significantly over 24 hours. Hence, the estimated difference is a quantitative
measure of our ability to correct for other temporally uncorrelated noise sources such
as orbit error and atmospheric propagation delay. In the case of our PS-InSAR
implementation, it is a measure of the ability of the StaMPS framework (Hooper,
2006) to distinguish the deformation signal from noise. We use the the standard
deviation of the difference in deformation estimates (σ (P (x, t+ 1)− P (x, t))) to
quantify the noise levels.
7.2.2 Inter-comparison with SBAS
Let VD (x) represent the difference in mean line-of-sight (LOS) velocities as estimated
by PS and SBAS techniques at point with spatial coordinates “x”. We analyze
the average difference, E (VD (x)) , and the standard deviation, σ (VD (x)), of the
difference in the estimated velocities over all commonly identified coherent pixels.
We calculate similar statistics for the estimated LOS time-series by assuming that
D (x, t) represents the difference in the estiamted PS and SBAS LOS deformation
time-series at point with spatial coordinates “x” at time instant “t”. In particular,
we estimate the average difference, E (D (x, t)), and the standard deviation of the
difference, σ (D (x, t)), by assuming that the difference of estimates at every time-
instance is independent. We also report the mean standard deviation E (σx (D (x, t))),
where σx represents the standard deviation for a fixed spatial index, between the
CHAPTER 7. NOISE IN PS-INSAR 119
time-series of commonly selected coherent pixels. Similar statistics have been used
in previous work, including the PSIC4 study (Raucoules et al., 2009), to validate the
estimated time-series against continuous GPS and/or leveling measurements (Casu
et al., 2006; Crosetto et al., 2008; Ferretti et al., 2009; Adam et al., 2009).
7.2.3 Comparison with alignment arrays
We compare the estimated creep across the Hayward Fault from PS-InSAR at 8
different stations with observations from the SFSU-USGS alignment array (McFar-
land et al., 2009). For estimating creep across the fault from the PS results, we
average the deformation estimates over an area of 500 m × 500 m on either side
of the fault adjacent to the station of interest and analyze the time-series of the
difference. We project the cumulative displacement observations of the creep meters
to radar LOS, assuming no verical deformation, for direct comparison with estimated
deformation time-series. We interpolate the creep meter observations using cubic
splines to create a time-series that corresponds to the SAR image acquisitions. We
represent the difference between the estimated PS time-series and the interpolated
creep meter measurements at a station x, as Δ(x, t) respectively. Like in Section 7.2.2,
we report the mean bias (E (Δ (x, t))) and the standard deviation (σ (Δ (x, t))) for the
8 alignment array stations. We also report the difference in estimated average velocity
(Δv (x)) between the PS-InSAR results and the alignment array creep measurements.
The SBAS results used in this work were previously compared against the alignment
array measurement and reported in Lanari et al. (2007a).
7.3 Results
Table 7.1 shows the statistics of the number of coherent pixels identified by the two
time-series InSAR techniques. We used a phase noise threshold of 5 mm in the
MLPS algorithm (Shanker and Zebker, 2007) for identifying the PS and a temporal
coherence threshold of 0.7 (see Pepe and Lanari (2006)) for identifying the SBAS
CHAPTER 7. NOISE IN PS-INSAR 120
pixels. The SBAS technique identified a slightly larger number of coherent pixels,
which is consistent with observations over other regions (Lauknes et al., 2010). Both
the techniques identified a similar network of coherent pixels (85 percent in common),
mostly in the urban areas. The MLPS algorithm performs slightly better in the non-
urbanized zones.
Table 7.1: The statistics of the number of coherent pixels identified by each time-series InSAR technique. The PS results were generated at single-look interferogramscale and the results were multi-looked to match the resolution of the SBAS results.
Property SFO Airport San Leandro South BaySize of the area in pixels(Azimuth × Range)
375 × 300 300 × 375 412 × 375
Pixels with PS estimates 13448 26956 55279Pixels with SBAS estimates 15653 29160 61510Number of common pixels 11222 23281 49134
Figure 7.3 shows the histogram of the difference in deformation estimates for one
of the tandem pairs (P (x, t+ 1)− P (x, t)) for the SFO airport region. It is evident
that the difference is normally distributed with almost zero mean. The non-zero bias
can be attributed to noise in the selected reference pixel. Table 7.2 tabulates the
tandem data noise levels for all the three tandem pairs and the sub-regions of the San
Francisco Bay Area and indicates that an average of 3 − 4 mm of noise is observed
between tandem pairs. Assuming that noise from both the scenes contribute equally
(factor of 1/√2) to this estimate, we estimate a noise level of 2−3 mm for every SAR
scene.
Table 7.2: Standard deviation of the difference [σ (P (x, t+ 1)− P (x, t))] in estimatedPS-InSAR time-series for the three tandem pairs of data in our series of 43 SAR scenes.
Dates SFO Airport San Leandro South Bay10− 11 Nov 1995 2.79 mm 2.94 mm 2.85 mm29− 30 Mar 1996 3.76 mm 3.72 mm 3.67 mm03− 04 May 1996 3.88 mm 3.82 mm 3.81 mm
CHAPTER 7. NOISE IN PS-INSAR 121
Figure 7.3: Histogram of the LOS time-series differences between the 10 − 11 Nov1995 tandem pair for all the PS pixels over the SFO airport region. Note that thehistogram has a Gaussian structure. The best fit Gaussian distribution is also shown.
In Figure 7.4 we present the mean deformation velocity maps computed using
the two techniques. It is clear that both the PS and the SBAS approaches identify
the major deformation features in the San Francisco Bay Area. These include the
subsidence of the SFO airport runway (SFO airport sub-area, Figure 7.4 (a)), the
uplift due to the San Leandro synform (Marlow et al., 1999; Schmidt et al., 2005),
rapid subsidence of many areas near the coast like the Bay Farm Island (San Leandro
sub-area, Figure 7.4 (b)), the differential uplift across the Silver creek Fault in
San Jose (South Bay sub-area, Figure 7.4 (c)), and the creep across the Hayward
Fault near Oakland and Fremont (Figure 7.4 (b,c)). In addition to comparing mean
deformation velocities, we also directly compared the estimated time-series for select
stations shown in Figure 7.4 (a3,b3,c3). The individual time-series for creep across the
Hayward Fault (Figure 7.5 (b5)) and differential subsidence across the Silver Creek
Fault (Figure 7.5 (c5)), were produced by averaging rectangular regions on either side
of their fault and computing their difference. Subsidence of regions along the San
Francisco Bay like Candle Stick Point (Figure 7.5 (a1)), Bay Farm Island (Figure 7.5
(b1)), Alameda Ferry (Figure 7.5 (b2)) and Shoreline Park (Figure 7.5 (c1)) are
also given. Few systematic differences between the results from the two techniques
are observed, as expected. From the difference between the estimated velocities
CHAPTER 7. NOISE IN PS-INSAR 122
(Figure 7.4), we note that the MLPS technique underestimates the subsidence, by
more than 2 mm/yr, of rapidly subsiding reclaimed areas located near the Bay,
such as the runway of the SFO airport (Figure 7.5 (a3)), the edge of Bay Farm
Island (Figure 7.5 (b1)) and the shoreline park (Figure 7.5 (c1)). This could be
a consequence of the spatial filtering effects of the StaMPS framework (Hooper,
2006). In the StaMPS framework, deformation is estimated from the unwrapped
data using a combination of a spatial low pass filter and a temporal low pass filter.
If the dimensions of the filter are large, some of the localized subsidence features are
smoothed out. This filtering however, does not affect our estimates of fault creep
as the physical dimensions of these features are significantly larger than those of the
applied filters. We also observe some boundary effects in the processing. The three PS
data sets were processed independently to reduce the workflow data volumes while the
whole frame was processed as a single data set for the SBAS analysis before subsets
were created for comparison. As a result, the PS pixels near the subset boundaries
are insufficiently constrained in the phase unwrapping step compared to the SBAS
data set. Hence, the differences between the velocity estimates near the edges are
more than 1 mm/yr higher than over other parts of the image. Moreover, note that
the time-series estimated using the PS and SBAS techniques physically represent the
motion of the dominant scatterer in the resolution element and the average motion of
the scatterers in the resolution element respectively. These values could be physically
different. Different phase unwrapping algorithms (Hooper and Zebker, 2007; Pepe
and Lanari, 2006) could be another source of difference between the two results.
Table 7.3 shows the statistics related to the difference in deformation estimates from
the two time-series InSAR techniques. From Figure 7.6, it is clear that the error
statistics are normally distributed and can hence, be characterized by their mean and
standard deviation. As described in Section 7.2.2, we also report the mean pixel-by-
pixel standard deviation as this statistic has been widely used in other studies for
quantifying estimation errors and differences. In summary, by taking into account
CHAPTER 7. NOISE IN PS-INSAR 123
Figure 7.4: LOS velocity in the three different regions of the San Francisco Bay Areaas estimated using the MLPS method (Left) and the SBAS method (Middle). Thecommon reference point is indicated by a black triangle in each of the images. (Right)The differences between the velocity estimates are also shown. The stations for whichindividual time-series are shown in Figure 7.5 are marked by circles. The areas acrossthe fault used to compute the creep across the Hayward Fault (b3) and the differentialsubsidence across the Silver Creek Fault (c3) are also shown.
for Table 7.3 values, we can assume that the average difference in estimated LOS
velocities is in the order of 1 mm/yr and the estimated LOS time-series is 5 mm.
Figure 7.7 shows the estimated creep using PS-InSAR and the observed creep
from the SFSU-USGS alignment array. Stations H39A (Figure 7.7(c)) and HENC
(figure 7.7(e)) show the most discrepancy between creep meter observations and
CHAPTER 7. NOISE IN PS-INSAR 124
Figure 7.5: Estimated LOS time-series for three different stations for each of thethree different regions in the San Francisco Bay Area using the MLPS method andthe SBAS method. The difference between the deformation estimates for each ofthe stations is also shown. b5 and c5 represents the average difference across theHayward and Silver Creek Faults, respectively, computed in the areas highlighted inFigure 7.4(b3) and Figure 7.4(c3).
Figure 7.6: (a) Histogram of velocity differences for the SFO airport region. (b)Histogram of the LOS time-series differences for the SFO airport region. (c)Histogram of the pixel-by-pixel std. dev of LOS time-series for the SFO airportregion. Note that the first two histograms have a Gaussian structure and the lastone, a Rayleigh distribution structure.
CHAPTER 7. NOISE IN PS-INSAR 125
Table 7.3: The statistics of the differences between LOS velocities and time-series asestimated using the MLPS and the SBAS techniques for the three sub-regions in theSan Francsico Bay Area.
Difference statistic Formula SFO Airport San Leandro South BayMean bias in velocity dif-ference (mm/yr)
E (VD (x)) 0.17 -0.04 -0.23
Std. dev. of velocitydifference (mm/yr)
σ (VD (x)) 1.18 0.76 0.95
Mean bias in time-seriesdifference (mm)
E (D (x, t)) -0.50 1.44 -1.06
Std. dev. of time-seriesdifference (mm)
σ (D (x, t)) 6.56 5.39 6.84
Mean of pixel-by-pixelstd. dev.
E (σx (D (x, t))) 4.33 3.61 4.72
Figure 7.7: PS-InSAR time-series and SFSU-USGS alignment array creep estimatesfor 8 selected stations along the Hayward Fault.
estimated time-series. PS-InSAR results clearly identify the change in creep rate
for station HPAL (Figure 7.7(h)) in mid-1997. Creep meters are insensitive to local
vertical deformation where as InSAR is most sensible to vertical deformation by a
factor of up to 3. Any local vertical deformation can significantly affect our PS-InSAR
CHAPTER 7. NOISE IN PS-INSAR 126
deformation estimates. Table 7.4 shows the difference statistics between the PS-
InSAR deformation estimates and the interpolated alignment array measurements for
8 stations along the Hayward Fault. From Table 7.4, we conclude that the discrepancy
between the estimated creep from the PS-InSAR time-series and alignment array
measurements is 1.5 mm in LOS displacement and 0.5 mm/yr in LOS velocity. These
value are similar to those reported by Lanari et al. (2007a).
Table 7.4: Mean and standard deviation of the difference between estimated PS-InSAR deformation time-series and observed creep time-series from the SFSU-USGSalignment array for 8 stations.
StationMean Bias Std. dev Velocity differenceE (Δ (x, t)) σ (Δ (x, t)) |Δv (x) |
HLSA 0.53 mm 1.27 mm 0.20 mm/yrHLNC -0.07 mm 1.29 mm 0.55 mm/yrH39A 1.36 mm 1.21 mm 0.54 mm/yrH73A 0.27 mm 1.60 mm 0.68 mm/yrHENC 1.78 mm 1.31 mm 0.28 mm/yrH167 1.25 mm 0.75 mm 0.34 mm/yrHROS 0.52 mm 1.18 mm 0.63 mm/yrHPAL 0.08 mm 0.79 mm 0.22 mm/yr
7.4 Conclusions
In this chapter, we have analyzed the noise characteristics of our PS-InSAR technique
by focusing on difference between tandem SAR pairs. We have also compared the
accuracy of the Stanford University’s MLPS selection algorithm in the StaMPS
framework against that of the IREA-CNR’s SBAS technique, which was one of the
original participants of the PSIC-4 study. As major result, we notice that both the PS
and SBAS techniques identify the salient deformation features in the Bay Area, with
a similar coverage. In addition, the final values obtained for the average difference
in estimated LOS velocity and time-series are consistent with those estimated by the
ESA PSIC-4 study (Raucoules et al., 2009), for their mining test site in Gardanne,
CHAPTER 7. NOISE IN PS-INSAR 127
France and by independent studies (Casu et al., 2006; Lanari et al., 2007a). We
also compared our PS-InSAR estimates with observed creep measurements from the
SFSU-USGS alignment array and found that our estimates were accurate within 1.5
mm LOS displacement and 0.5 mm/yr LOS velocity, similar to the values reported
by Lanari et al. (2007a) for SBAS estimates.
Future comparative time-series InSAR studies should also include detailed
comparisons of recently developed algorithms, that estimates deformation at the
highest possible resolution from short baseline interferograms (Hooper, 2008; Lanari
et al., 2004b). Similar comparative studies are needed for data acquired using X-band
and L-band sensors to better understand the effects of wavelength, resolution and
orbit repeat periods on the ability of time-series InSAR algorithms to study various
geophysical phenomena. The ESA supersite initiative will enable us to conduct more
inter-comparative studies and validate them against other geodetic measurements and
in the process, will enable us better understand the limitations of these time-series
techniques.
Chapter 8
Summary
Applicability of conventional interferometric synthetic aperture radar (InSAR) for
crustal deformation studies is limited by the fact that almost any interferogram
includes large areas where the signals decorrelate and no reliable measurement is
possible. Persistent scatterer (PS) InSAR overcomes the decorrelation problem by
identifying resolution elements whose echo is dominated by a single scatterer in a
series of interferograms. Combining information from multiple interferograms also
mitigates the effect of atmospheric distortions in the deformation estimates. Existing
PS methods have been very successful in analysis of urban areas, where man-made
structures act like strong corner-reflectors. However, man-made structures are absent
from most of the Earth’s surface. The Stanford method for persistent scatterers
(StaMPS) was the first technique designed to extend the applicability of PS-InSAR
to natural terrain.
This dissertation has three major aspects to it - i) improving PS selection in
natural terrain, ii) developing new phase unwrapping algorithms suitable for multi-
dimensional time-series InSAR datasets and iii) describing the noise characteristics of
the deformation estimated using PS-InSAR. In Chapter 4 we introduce an information
theoretic approach to PS selection that identifies a denser PS network in natural
terrain than other published algorithms.
In Chapter 5 we address the phase unwrapping problem and presents two new
unwrapping algorithms designed for time-series InSAR applications. The second of
128
CHAPTER 8. SUMMARY 129
these techniques, the “edgelist” algorithm, relies on a new robust phase unwrapping
formulation that is more flexible than the conventional network programming
formulations and can incorporate additional geodetic observations as constraints.
In Chapters 6 and 7, we apply our PS-InSAR technique to study rockslides in
Lyngen region of Norway and crustal deformation in the San Francisco Bay Area
respectively. We also characterize the noise levels in our deformation estimates by
comparing our estimates against independent SBAS estimates from research groups
in Europe. We observe that our deformation estimates are within 1 mm/yr LOS
velocity and 5 mm absolute LOS displacement of the SBAS results. When compared
against creep measurements from alignment arrays along the Hayward fault our PS-
InSAR time-series matched ground reality to within 1.5 mm LOS displacement and
0.5 mm/yr LOS displacement respectively.
8.1 Future Work and Improvements
PS selection in this work use statistics derived from assumed signal models for the
scattering behavior of PS pixels. However, very little work has been done to observe
and verify the actual scattering characteristics of PS pixels. Understanding the
effect of wavelength, pixel resolution and frequency of SAR acquisitions will help
us understand the scattering properties of PS pixels better and will provide insight
into the actual scattering mechanisms at work. Such information will be crucial
for designing PS selection techniques that will perform better in non-urban terrain.
Understanding of these aspects of PS-InSAR is also crucial for design and development
of future SAR sensors.
Extracting deformation signatures from the idenfied PS network using phase
unwrapping algorithms remains one of the harder aspects of time-series InSAR
processing. We have developed the “edgelist” algorithm during the course of this
dissertation but all aspects of this formulation have not yet been completely explored.
In particular,
CHAPTER 8. SUMMARY 130
1. Fast implementation of this algorithm for large scale time-series InSAR
problems with efficient customized linear solvers is yet to be explored.
2. The possibility of using softer inequality constraints as opposed to hard equality
constraints using additional geodetic data is yet to be explored.
3. Adjustment of cost functions using a priori information regarding the area of
study needs to be formalized. We demonstrated in Chapter 5, that adjustment
of cost functions is an important issue that needs to be addressed for correctly
unwrapping time-series InSAR data across the San Andreas Fault.
Over the last few years, time-series InSAR has taken over from conventional InSAR
as the default technique for crustal deformation studies. With the launch of several
SAR missions like TerraSAR-X, ALOS PALSAR, COSMO-SkyMed etc, SAR data is
being acquired more frequently and in larger volumes than ever before. For maximum
utilization of this data resource, automation of time-series InSAR techniques is the
need of the hour.
8.2 Conclusions
We have demonstrated the applicability of PS-InSAR techniques in natural terrain in
Chapters 4, 5, 6 and 7. We have demonstrated the use of our maximum likelihood PS
selection technique and the edgelist unwrapping technique to study time-dependent
creep across the Central San Andreas in Chapter 5. We also demonstrate the use
of PS-InSAR techniques for studying rockslide activity in Norway in Chapter 6. We
have compared our deformation estimates against the SBAS results from research
groups in Europe in Chapters 6 and 7, and conclude that our results agree to within
1 mm/yr LOS velocity and 5 mm absolute LOS displacement. Our deformation
estimates matched to within 1.5 mm LOS displacment and 0.5 mm/yr LOS velocity
when compared against creep measurements from alignment arrays along the Hayward
Fault.
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